Phy 116 Lab Manual Fall 15

7/24/2019 Phy 116 Lab Manual Fall 15

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PHYS

ICSLABM

ANU

AL

E

NGINEERING

SCIENCE&P

HYSICS

DEPARTMENT

C O L L E G E O F S T T E N I S L N D

C I T Y U N I V E R S I T Y O F N E W Y O R K

PHY 116

The important thing in science is not so much to obtain newfacts as to discover new ways of thinking about them.

--Sir William Lawrence Bragg


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COLLEGE OF STATEN ISLAND

PHY 116

PHYSICS LAB MANUAL

ENGINEERING SCIENCE & PHYSICS DEPARTMENT

CITY UNIVERSITY OF NEW YORK


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PHYSICS LABORATORYEXT 2978, 4N-214/4N-215



LABORATORY RULES

1. No eating or drinking in the laboratory premises.

2. The use of cell phones is not permitted.

3. Computers are for experiment use only. No web surfing, reading e-mail,instant messaging or computer games allowed.

4.

When finished using a computer log-off and put your keyboard and mouse

away.

5. Arrive on time otherwise equipment on your station will be removed.

6. Bring a scientific calculator for each laboratory session.

7. Have a hard copy of your laboratory report ready to submit before you enterthe laboratory.

8.

Some equipment will be required to be signed out and checked back in. The

rest of the equipment should be returned as directed by the technician. Allequipment must be treated with care and caution. No markings or writing is

allowed on any piece of equipment or tables. Remember, you are responsiblefor the equipment you use during an experiment.

9.

After completing the experiment and, if needed, putting away equipment,

check that your station is clean and clutter free and push in your chair.

10. Before leaving the laboratory premises, make sure that you have all yourbelongings with you. The lab is not responsible for any lost items.

Your cooperation in abiding by these rules will be highly appreciated.

Thank You.

The Physics Laboratory Staff


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10 ESSENTIALS of

writing laboratory reports ALL students must comply with

1.No report is accepted from a student who didnt actually participate in the

experiment.

2.

Despite that the actual lab is performed in a group, a report must be individually

written. Photocopies or plagiaristic reports will not be accepted and zero grade

will be issued to all parties.

3. The laboratory report should have a title page giving the name and number ofthe experiment, the student's name, the class, and the date of the experiment.

The laboratory partners name must be included on the title page, and it should

be clearly indicated who the author and who the partner is.

4. Each section of the report, that is, objective, theory background, etc.,should be clearly labeled. The data sheet collected by the author of the report

during the lab session with instructors signature must be included no report

without such a data sheet indicating that the author has actually performed the

experiment is to be accepted.

5. Paper should be 8 x 11. Write on one side only using word-processing

software. Ruler and compass should be used for diagrams. Computer graphingis also accepted.

6. Reports should be stapled together.

7. Be as neat as possible in order to facilitate reading your report.

8. Reports are due one week following the experiment. No reports will beaccepted after the "Due-date" without penalty as determined by the instructor.

9. No student can pass the course unless he or she has turned in a set of laboratory

reports required by the instructor.

10. The student is responsible for any further instruction given by the instructor.


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PHY 116

TABLE OF CONTENTS

The laboratory instructor, in order to adjust to the lecture schedule or personal preference, maysubstitute any of the experiments below with supplementary experiments.

1. LABORATORY REPORT FORMAT AND DATA ANALYSIS........................................1

2. VERNIER CALIPER - MICROMETER CALIPER............................................................9

3. MASS, MASS DENSITY, SPECIFIC GRAVITY..............................................................15

4. ADDITION OF VECTORS................................................................................................21

5. MOTION OF A BODY IN FREE FALL............................................................................25

6. HORIZONTAL PROJECTILE MOTION...........................................................................31

7. EQUILIBRIUM OF A RIGID BODY.................................................................................33

8. FRICTION...........................................................................................................................35

9. NEWTONS SECOND LAW.............................................................................................39

10. SIMPLE PENDULUM.........................................................................................................43

11. CENTRIPETAL FORCE....................................................................................................45

12. WORK AND KINETIC ENERGY......................................................................................47

13. CONSERVATION OF MOMENTUM................................................................................49

14. ROTATIONAL MOTION AND MOMENT OF INERTIA.................................................51

SUPPLEMENTARY EXPERIMENTS

15. DENSITY AND ARCHIMEDES PRINCIPLE...................................................................55

16. COLLISION IN TWO DIMENSIONS................................................................................59

17. VIBRATIONS OF A SPRING................................................................................................61

18. CALORIMETRY.................................................................................................................63

19. EQUILIBRIUM AND CENTER OF MASS.........................................................................65


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20. ATWOODS MACHINE........................................................................................................67

21. SOUND WAVES................................................................................................................71

APPENDIX:

A1 GRAPHICAL ANALYSIS 3.4 - FINDING THE BEST FIT.................................................77

All diagrams and tables created by Jackeline S. Figueroa, Senior CLT.Except for diagrams on pages 9-12, 21 and 77-80


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LABORATORY REPORTS FORMAT AND PRESENTATION OF DATA

The Laboratory Report should contain the following information:

1. Objective of the lab;

2. Physical Principles and laws tested and used;

3. Explanation (rather than a description) of the procedure;

4. Laboratory Data: arranged in tabular form with labeled rows and columns. Note that

the data sheet must be signed by the instructor in the presence of the student whenthe experiment is completed;

5. Computations and graphs of the main quantities and their errors;

6. Summary of Results which includes: discussion of the results and their errors as well asa conclusion based on this discussion as to what extent the lab objective is achieved.

7. Answers to all questions.

I. ERRORS OF OBSERVATION

1. Blunders:

Every measurement is subject to error. Obviously, one should know how to

reduce or minimize error as much as possible. The commonest and simplest type of errorto remove is a blunder, due to carelessness, in making a measurement. Blunders are

diminished by experience and the repetition of observations.

2. Personal Errors:

These are errors peculiar to a particular observer. For example

beginners very often try to fit measurement to some preconceived notion. Also, the

beginner is often prejudiced in favor of his first observation.

3. Systematic Errors:

Are errors associated with the particular instruments or technique of

measurement being used. Suppose we have a book which is 9in. high. We measure itsheight by laying a ruler against it, with one end of the ruler at the top end of the book. If

the first inch of the ruler has been previously cut off, then the ruler is like to tell us that

the book is 10in. long. This is a systematic error. If a thermometer immersed in boiling

pure water at normal pressure reads 215F (should be 212F) it is improperly calibrated;if readings from this thermometer are incorporated into experimental results, a systematic

error results.

4. Accidental (or Random) Errors: When measurements are reasonably free from the

above sources of error it is found that whenever an instrument is used to the limit its

precision, errors occur which cannot be eliminated completely. Such errors are due to thefact that conditions are continually varying imperceptibly. These errors are largely

unpredictable and unknown. For example: A small error in judgment on the part of the

observer, such as in estimating to tenths the smallest scale divisions. Other causes are


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unpredictable fluctuations in conditions, such as temperature, illumination, socket voltage

or any kind of mechanical vibrations of the equipment being used. The effect of these

errors may be mitigated by repeating the measurement several times and taking theaverage of the readings.

There are two ways of estimating the error due to random independent measurements.

One way is to calculate the Mean Absolute Deviation and the other is to calculate theStandard deviation. Both methods are discussed in the appendix.

5. Significant Figures:

Every number expressing the result of a measurement or of

computations based on measurements should be written with the proper number of"significant figures." The number of significant figures is independent of the position of

the decimal point: i.e. 8.448cm, 84.48mm, or 0.08448m has the same number of

significant figures. A figure ceased to be a significant when we have no reason to believe,on the basis of measurement made, that the correct result is probably closer to that figure

than to the next (higher or lower) figure. In computations, since figures which are not

significant in this sense have no place in the final result, they should be dropped to avoid

useless labor. e.g. in the measurement of the diameter of a penny we read on the ruler1.748. Here the last figure is a very rough guess; hence, for computations we use 1.75.

6. Reading error

: Every instrument has a limitation in accuracy. The markings serve as aguide as to that accuracy. We read the instrument to a fraction of the smallest division. As

in the diameter of a penny the 8 is an estimated number. We then have to estimate the

error in that number. For most applications the reading error can be taken as +/-2.Therefore the experimental value for that measurement is 1.748 +/- 0.002 cm. The

reading error may be taken as a constant error for that instrument. The smallest error

associated with a measurement is the reading error.

7. Percent Error

The Measurement error may be estimated from your measurements a variety of ways.Two simple ways are the standard deviation or the mean absolute error. For most

applications the mean absolute error is a good estimate of the measurement error.

: To present the error in a relative manner we calculate the percent error.

Percent Error =|Measurement Error|

Average Value 100%

8. Percent Difference:

Percent Difference =

|Standard Value Experimental Value|

Standard Value 100%

In your laboratory work you will often find occasion to compare a

value which you have obtained as a result of measurement, with the standard, orgenerally accepted value. The percent difference is computed as follows:

Note: If Percent Difference (PD) is smaller than Percent Error (PE), you can concludethat the experimental value is consistent with the standard (known) value within

estimated errors. If, however, PD is larger than PE, the measured value is inconsistent

with the standard (known) one. In other words, if PE is estimated correctly, the measuredvalue can be claimed to be a better estimate of the standard one.


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II. ANALYSIS OF DATA

Every measurement is prone to errors leading to deviationsof a measured quantity fromone measurement to another. For example, length of a pencil measured several times may come

out differently depending on how ruler was applied. Personal blunders due to carelessness are

also a source of errors. In general, each particular instrument never gives a result precisely. Manyexternal factors such as, e.g., temperature vary and thereby affect results. Thus, errors of

measurements and the associated deviations of measured quantity are an inherent part of the

measurement process. Patience and experience are required in order to reduce the errors and the

deviations.

In order to evaluate errors the same quantity should be measured at least several times.

As an example, the result of such measurements of a length of one object is given in the table

below

N 1 2 3 4 5 6 7 8 9

L

[cm]15.2 15.3 14.9 15.4 15.2 15.1 15.0 14.8 15.2

DL=L - L [cm]

0.1 0.2 0.2 0.3 0.1 0.0 0.1 0.3 0.1

The upper row marked by N gives the number of a particular measurement. The second row

shows objects length obtained during each measurement (for example, the result of the 4thmeasurement is 15.4 cm). The bottom row gives absolutedeviations

|LL|DL = Eq. 1

of each measurement from the average value (mean value) of the length

cm1.159

)2.158.140.151.152.154.159.143.152.15()L(avgL =

++++++++==

calculated from 9 measurements. In calculating the average, the result must be rounded off so

that the number of significant digits is not more than that for each measurement. The mean

absolute deviation

DL avg DL= ( ) Eq. 2

indicates how the measured value varied due to all of the factors mentioned above. For our

example, D L=0 2. cm. The final result for the object length is expressed as

L L DL= Eq. 3

That is, L cm= ( . . )151 02 . This means that in the measurement of the length the result obtainedwas between 14.9 cm and 15.3 cm with high certainty.


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Errors can also be represented aspercent error. It is defined as

%100L

LDerror% = Eq. 4

For our example, it is

0 2

151 100% 1%

.

. . This sort of analysis should be applied tomeasurements of other physical quantities.

Sometimes a purpose of the laboratory experiment is to measure a quantity Q whose

standard value Qst is well known from theoretical considerations or other measurements. In thiscase it is important to comparethese two quantities Q and Qst in order to make a conclusion on

whether your experiment confirmed the value Qst and thereby supported a theoretical concept

underlying this value. An important quantity is the percent difference between the measured

(mean) value and thestandardvalue:

| |

% difference 100%

st

st

Q Q

Q

=

Eq. 5

We can say that

The errors should always be estimated for the experimental data. Furthermore, any

experimental result for which no errors are evaluated is considered as unreliable.

the experiment does confirm the concept within the experimental percent

deviation(or percent error), if the percent dif ference is not bigger than the percent error.

PROPAGATION OF ERRORS

Sometimes a measured quantity is obtained by using some equation, and the question is how toevaluate fractional (or percent) error for such a quantity. For example, density of some material

is obtained as the ratio of mass M and volume V: =M/V. While mass can be measured directly

by scale, volume is often obtained from measurements of linear dimensions of a rectangularshaped sample as V=L1L2L3. Each four values, M, L1, L2, and L3have their own errors (mean

deviations):

The resulting fractional (or % ) error for can be found as a sum of fractional (%) errors ofall multiplicative quantities entering the equation.

For our example this means

31 2

1 2 3 1 2 3

,LL LM M

M L L L L L L

= + + + = Eq. 7

M = M ML1 = 1 1L2 = 2 2

L3 = 3 3

Eq. 6


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Let us use particular measurements performed on a piece of wood of mass M with rectangular

shape given by dimensions L1,L2andL3:

Then, the mean volume is 2.4.2.0

.3.4=16 cm

3and the mean density becomes:

3 7.5 /16 0.47 cmg

= =

The fractional error follows from Eq. 7 as

0.2 0.1 0.1 0.1 0.15

7.5 2.4 2.0 3.4

= + + + =

That is, % error is 0.15.

100%=15%, and the absolute error is 0.47.

0.15 g/cm3

=0.07 g/cm3

.

The final answer for the density is

3 (0.47 0.07) cmg

=

Similar procedure should be followed for other composite quantities.

STANDARD DEVIATION

The method of estimating errors as the mean of the deviations shown in Eq. 2 to Eq. 4 can be

improved by considering these deviations as some random variable. Then, thestandard deviationof such variable from its mean is taken as the error. In general, the procedure becomes as

follows:

If the random variable X takes on N values x1,,xN (which are real numbers) with equal

probability, then its standard deviation can be calculated as follows:

1. Find the mean, x , of the values.

2. For each value xi, calculate its deviation )xx( i from the mean.

3. Calculate the squares of these deviations.4. Find the mean of the squared deviations. This quantity is the variance

2.

5. Take the square root of the variance.

6. This calculation is described by the following formula:

M = (7.50.2)gL1 = (2.40.1)cm

L2 = (2.00.1)cmL3 = (3.40.1)cm

=

=N

1i

2

i )xx(N

1


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where x is the arithmetic mean of the values xi, defined as:

Example:

Suppose we wished to find the standard deviation of the data set consisting of the values 3, 7, 7,

and 19

Step 1:Find the arithmetic mean (average) of 3, 7, 7, and 19,

94

19773=

+++

Step 2:Find the deviation of each number from the mean,

3 9 = -67 9 = -2

7 9 = -2

19 9 = 10

Step 3: Square each of the deviations, which amplifies large deviations and makes negative

values positive,(-6)

2 = 36

(-2)2 = 4

(-2)2 = 4

102 = 100

Step 4:Find the mean of those squared deviations,

Step 5: Take the non-negative square root of the quotient (converting squared units back to

regular units),

So, the standard deviation of the set is 6. This example also shows that, in general, the standarddeviation is different from the mean absolute deviation, as calculated in Eq. 2. Specifically for

this example the mean deviation is 5. Despite these differences, both methods of estimatingerrors are acceptable.

=

=+++

=N

1ii

N21 xN

1

N

x...xxx

364

1004436=

+++

636 =


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III. GRAPHICAL REPRESENTATION OF DATA: Some essentials in plotting a graph.

1. Arrange the numbers to be plotted in tabular form if they are not already so arranged.

2. Decide which of the two quantities is to be plotted along the X-axis and which along the

Y-axis. It is customary to plot the independent variable along the X-axis and thedependent along the Y.

3. Choose the scale of units for each axis of the graph. That is, decide how many squares

of the cross-section plotted along a particular axis. In every case the scales of units forthe axes must be so chosen that the completed curve will spread over at least one-half of

the full-sized sheet of graph paper.

4. Attach a legend to each axis which will indicate what is plotted along that axis and, in

addition, mark the main divisions of each axis in units of the quantity being plotted.

5. Plot each point by indicating its position by a small pencil dot. Then draw a small circlearound the dot so that each plotted point will be clearly visible on the completed graph.

This circle is drawn with a radius equal to the estimated probable error of that particular

measurement (you may use the percent difference when calculable). (See "errors"below).

6. Draw a smooth curve through the plotted points. This curve need not necessarily passthrough any of the points but should have, on the average, as many points on one side of

it as it has on the other. The reason for drawing a smooth curve is that it is expected to

represent a mathematical relationship between the quantities plotted. Such amathematical relationship ordinarily will not exhibit any abrupt changes in slope, merely

indicates that the measurement is subject to some error. A close fit of the experimental

points to the smooth curve shows that the measurement is one of small error.

7. Label the graph. That is, attach a legend which will indicate, at a glance, what the graph

purports to show.


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Fig. 1 - Parts of a Vernier Caliper

Fig. 2 - Dimensions that can be measured with a vernier caliper

ENGLISH SCALE

VERNIER CALIPER - MICROMETER CALIPER

Apparatus:

- Two metal cylinders

- One wire- Vernier caliper, 0-150mm, 0.02 least count

- Micrometer caliper, 0-25mm, 0.01mm least count

Part I: The Vernier Caliper

When you use English and metric rulers for making measurements it is sometimes difficult to get

precise results. When it is necessary to make more precise linear measurements, you must have a

more precise instrument. One such instrument is the vernier caliper.

The vernier caliper was introduced in 1631 by Pierre Vernier. It utilizes two graduated scales: a

main scale similar to that of a ruler and a especially graduated auxiliary scale, the vernier, that slides

parallel to the main scale and enables readings to be made to a fraction of a division on the mainscale. With this device you can take inside, outside, and depth measurements. Some vernier calipers

have two metric scales and two English scales. Others might have the metric scales only.

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Fig. 3 - Vernier caliper with closed jaws

Notice that if the jaws are closed, the first line at the left end of the vernier, called the zero line or

the index, coincides with the zero line on the main scale (Fig. 2).

The least count can be determined for any type of vernier instrument by dividing the smallestdivision on the main scale by the number of divisions on the vernier scale. The vernier caliper to

be used in the laboratory measurements has a least count 0.02mm. Instructions on how to read the

measurements on this particular model can be found in:

http://www.chicagobrand.com/help/vernier.html

http://www.technologystudent.com/equip1/vernier3.htm

The link below has a caliper simulator, practice with it before the lab session:

http://www.stefanelli.eng.br/en/en-vernier-caliper-pachymeter-calliper-simulator-millimeter-02-

mm.html

For our experiment will be using a caliper with English and Metric scales. The top main scale is

English units and the lower main scale is metric. For our experiment will be concentrating on metric

only. In our model the metric scale is graduated in mm and labeled in cm. That is, each bar

graduation on the main scale is 1mm. Every 10thgraduation is numbered (10mm). The vernier scale

divides the millimeter by fifty (1/50), marking the 0.02mm (two hundredths of a millimeter), which

is then the least count of the instrument. In other words, each vernier graduation corresponds to

0.02mm. Every 5thgraduation (0.1mm) is numbered.

Having first determined the least count of the instrument, a measurement may be made by closing

the jaws on the object to be measured and then reading the position where the zero line of the vernier

falls on the main scale (no attempt being made to estimate to a fraction of a main scale division). We

next note which line on the vernier coincides with a line on the main scale and multiply the number

represented by this line (e.g., 0,1,2, etc.) by the least count on the instrument. The product is then

added to the number already obtained from the main scale. Occasionally, it will be found that no line

on the vernier will coincide with a line on the main scale. Then the average of the two closest lines

is used yielding a reading error of approximately 0.01mm. In this case we take the line that most

coincides.

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Fig. 4 - Sample reading on vernier caliper

Let us review the steps on how to use a vernier caliper (Fig. 4), note that we are only interested in

metric measurements. Before taking a measurement make sure the vernier reads zero when the jaws

are fully closed. If this is not the case, request another caliper, as it could be damaged.

Step 1: The main metric scale is read first. In our example there are 21 whole divisions (21mm)

before the 0 line on the vernier scale. Therefore, the first number is 21.

Step 2: On the vernier scale, find the first line that lines up with any line on the main scale. This is

shown by the arrow pointing in the example (lower vernier scale) to the 16thline.

Step 3: Multiply 16 by the least count 0.02, thus resulting in 0.32 (remember, each division on the

hundredths scale (vernier scale) is equivalent to 0.02mm. Thus, 16 x 0.02=0.32mm.

Step 4: Add 21 and 0.32, that is, 21+0.32=21.32mm. Thus, your final reading is 21.320.01mm.

Alternatively, it is just as easy to read the 21mm on the main scale and 32 on the hundredths scale,

therefore resulting in 21.32 as your measurement. That is, 21.320.01mm.

Procedure:

1. Make six independent measurements of the diameter of each metal cylinder.

2. Make six independent measurements of the length of each metal cylinder

3. Make six independent measurements of the diameter of the wire.

Questions:

1. Why did you take six independent measurements in each procedure above?

2. What does the smallest division on the main scale of the vernier caliper correspond

to?

3. What is the error of your measurements?

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Fig. 5 - Micrometer Caliper

Part II. The Micrometer Caliper:

The micrometer caliper, invented by William Gascoigne in the 17thcentury, is typically used to

measure very small thicknesses and diameters of wires and spheres. It consists of a screw of pitch

0.5mm, a main scale and another scale engraved around a thimble which rotates with the screw and

moves along the scale on the barrel. The barrel scale is divided into millimeters, on someinstruments, such as ours, a supplementary scale shows half millimeters.

The thimble scale has 50 divisions. Since one complete turn of the thimble will produce an axial

movement of 0.5mm. One scale division movement of the thimble past the axial line of the scale on

the barrel is equivalent to 1/100 times 1.0 equals 0.01mm. Hence readings may be taken directly to

one hundredth of a millimeter and by estimation (of tenths of a thimble scale division) to a

thousandth of a millimeter.

The object to be measured is inserted between the end of the screw (the spindle) and the anvil on the

other leg of the frame. The thimble is then rotated until the object is gripped gently. A ratchet at the

end of the thimble serves to close the screw on the object with a light and constant force. Thebeginner should always use the ratchet when making a measurement in order to avoid too great a

force and possible damage to the instrument.

The measurement is made by noting the position of the edge of the thimble on the barrel scale and

the position of the axial line of the barrel on the thimble scale and adding the two readings. The

micrometer should always be checked for a zero error. This is done by rotating the screw until it

comes in contact with the anvil (use the ratchet) and then noting whether the reading on the thimble

scale is exactly zero. If it is not, then this "zero error" must be allowed for in all readings.

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Fig. 6 - Sample Reading on Micrometer

To read a measurement (Fig. 6), simply add the number of half-millimeters to the number of

hundredths of millimeters. In the example below, we have 2.6200.005mm, that is 5 half

millimeters and 12 hundredths of a millimeter. If two adjacent tick marks on the moving barrel look

equally aligned with the reading line on the fixed barrel, then the reading is half way between the two

marks.

In the example above, if the 12 thand 13thtick marks on the moving barrel looked to be equally

aligned, then the reading would be 2.6250.005mm.

You may use this java applet to practice the use and reading of a micrometer.

http://www.stefanelli.eng.br/en/micrometer-caliper-outside-millimeter-hundredth.html

Procedure:

1. Repeat all measurements that are possible of part I (vernier caliper) using the micrometer.

2 Make six independent measurements of the diameter of a human hair.

3. What is the error of your measurements?

Questions:

1. Would you use the vernier to measure the diameter of a human hair? Explain your answer.

2. What does one division on the barrel of the caliper correspond to?

3. What does one division on the rotating thimble correspond to?

4. Define metric scale.

5. What does pitch 0.5 mm mean?

6. What type (name) of error is the "zero error" of the micrometer assuming it enters a calculation

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MASS, MASS DENSITY, AND SPECIFIC GRAVITY

Apparatus:

- Electronic balance- Vernier caliper

- Micrometer caliper

- Assorted metallic cylinders

- Aluminum bar

- Wooden block

- Irregular shaped object (mineral sample)

- 250ml graduated cylinder

Part I. Mass and Weight:

The mass of a body at rest is an invariable property of that body. It is a measure of the quantity ofmatter in a body. A body has the same mass at the equator as at the North Pole, -- the same mass

on the earth as on Jupiter or interstellar space.

The gravitational force between the earth (or other planet) and a body is called the weight of the body

with respect to the earth (or other planet). The gravitational force on a body is a variable quantity

even on the surface of the earth, e.g., the weight of a body is larger at the North Pole than at the

Equator. E.g., A book transported to the moon would have the same mass (quantity matter) on the

moon as it had back on the earth, but the book weighs less on the moon than it did on the earth

because the moon's gravitational pull is less than the earth's.

The weight of a body is proportional to its mass, the proportionality factor depending on the place

at which the weight is determined. If the weight of a body is compared with that of a standard body,

at the same place on the earth the ratio of the two weights is equal to the ratio of the two masses.

Consequently, if the weight of the body is found to be equal to the weight of a standard body at the

same place on earth, the two masses are equal. In order to measure the mass of a body, it is

necessary to find a standard mass or a combination of standard masses whose weight equals that of

a body at the same place on the earth. The device employed for this purpose is called a balance.

Procedure:

1.1. Determine the mass of each object using the balance. Record all data in tabular form.A suggested format for the cylinders and wires is shown:

ObjectUsed

Mass[g]

Diameter[cm]

Height[cm]

Volume Density

AbsoluteError

S.G.FromCalipers

[cm3]

FromDisplacement

[cm3]

FromCalipers[g/cm3]

Displacement[g/cm3]

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Fig. 1 - Volumes of a Cylinder and a Block

Design your own table for the aluminum bar and wooden block. Think of the dimensions you

are measuring in this case and that would help you determine what columns you would need

on your new table.

Part II. Volumes by measuring dimensions:

Procedure:

2.1. Using the vernier and micrometer calipers, make the necessary measurements to enable you

to calculate the volume of the regular bodies. Repeat each measurement at least once and take

the average.

Part III. Measuring the volume with the graduated cylinder:

The graduated cylinder used to measure the volume of a liquid has the scale in milliliters. A liter

is a unit of volume used in the metric system. There are 3.79 liters to a U.S. gallon, but for our

purposes:

1 Liter = 1000 ml = 1000 cubic centimeters (cm3or cc)

or more usefully:

1ml=1cc

If one pours water into a graduated cylinder one notices the top surface of the water is curved (Fig.

2). The curved surface is called a meniscus. The curvature is due to cohesive forces between the

inner wall for the graduated cylinder and the water in contact with it. We read the column of water

by looking at the correspondence of the bottom of the meniscus with the scale of the cylinder.

It was Archimedes who noted that any object of any shape when placed in a liquid displaced its own

volume. Thus, placing an object in our graduated cylinder (which now contains some water) we note

that the water level rises.

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Fig. 2 - Graduated Cylinder

Procedure:

3.1 Use the graduated cylinder to obtain the volume of the objects applicable to this method. Be

ingenious with the wooden block!

Part IV. Mass Density and Weight Density:

The mass density of a material is defined as the mass of any amount of that material divided by

the volume of that amount. The density of a substance is a fixed quantity for fixed external

conditions, and, thus, is a means of identifying a substance. e.g., All different shaped solids of

aluminum have the same density at room temperature. The units of mass density are g/cm3or kg/m3

in the metric system.

When we use centimeter (cm), grams (g), and seconds (s) in measuring quantities we refer it as the

cgssystem. Likewise when we use meters (m), kilograms (kg), and seconds (s) we refer to it as the

mkssystem.

In the English system mass is measured in the unit slug. Note that 1 slug is equal to 14.59 kg.

Therefore, the mass density in the English system may be expressed as slugs/ft3.

Water has a mass density of 1.94 slug/ft3in the English, and 1 g/cm3in cgs.

Procedure:

4.1. Calculate the mass density of each object in the cgs system.

4.2. Convert all your densities to the English system.

4.3. Identify the unknown object(s) by using the density(ies) you calculated and finding a close

match in the Density Table shown below:

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Table of Densities of Common Substances: See the American Institute of Physics Handbook for

a more extensive list. All values in cgs (g/cc) and at 20C.

NameDensity

[g/cm3]Name

Density

[g/cm3]Name

Density

[g/cm3] Name

Density

[g/cm3]

Aluminum 2.70 Calcite 2.72 Ash 0.56 Cement 1.85

Brass 8.44 Diamond 3.52 Balsa 0.17 Chalk 1.90

Copper 8.95 Feldspar 2.57 Cedar, red 0.34 Clay 1.80

Iron 7.86 Halite 2.12 Corkwood 0.21 Cork 0.24

Lead 11.48 Magnetite 5.18 Douglas Fir 0.45 Glass 2.60

Nickel 8.80 Olivine 3.32 Ebony 0.98 Ice 0.92

Silver 10.49 Mahogany 0.54 Sugar 1.59

Tin 7.10 Oak, red 0.66 Talc 2.75

Zinc 6.92 Pine, white 0.38

4.4 Calculate the % difference of your density measurements.

Part V. Relative Density or Specific Gravity (S.G.)

Because the number expressing the density depends on which units are used it is often advantageous

to be able to state a density in such a way that the number is independent of the system of units. We

can do this by giving the relative density, that is, the number of times the substance is denser than

water. The relative density is called the specific gravity (S.G.). In the form of the equation:

WhereDis the density of the substance, andDw is the density of water.

Later in the term you will see that if a substances S.G.is less than 1.0it floats in water and if it is

greater it sinks.

As an example ofDof iron in cgsis 7.8 g/cm3andDis 1.0therefore the S.G. of iron is:

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In the English system theDof iron is 15.1 slugs/ft3andDwis 1.94 slug/ft3therefore

Procedure:

5.1 Use the densities in the cgs system you obtained and calculate the S.G.of each substance.

5.2 Use your English figures for the densities and calculate the S.G.

Questions:

1. By Archimedes' observation how would you obtain the volume of the object placed in the

cylinder?

3. Which value of the volume is closer to the 'truth'? i.e., Part II or III. Explain your answer.

4. How do you account for the errors in your computed values of the density(ies)?

5. Which type of measurement done in Parts I, II and III do you think you made with the least

error? i.e., mass or length or volume. Explain.

6. Which of the densities you determined would you expect to be the least accurate?

7. Would you expect that the density of the wires would be as accurate as the value obtained fora cylinder of the same material?

8. Why do you think the densities would change if you changed the temperature?

9. What is the benefit, if any, in measuring volumes by using Archimedes observations?

10. In the above calculations of the S.G. in the Metric and English system what observations can

you make about the S.G.?

11. Estimate errors of your measurements in each procedure.

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Fig. 1 - Set-up of Force Table

Fig. 2 - Resultant and Equilibrant for

Two Forces

ADDITION OF VECTORS

Apparatus:

SForce table

S Four pulleys

S Four weight hangers

S Slotted weights

S Level

S Protractor

S Metric Ruler

S Graph paper

Introduction:

When a system of forces, all of which pass through the same point, acts on a body, they may bereplaced by a single force called the resultant. The purpose of this experiment is to show that the

magnitude and direction of the resultant of several forces acting on a particle may be determined by

drawing the proper vector diagram, and that the particle is in equilibrium when the resultant force

is zero.

The apparatus used in this experiment (Fig. 1) consists of a horizontal force table graduated in

degrees and provided with pulleys which may be set at any desired angle. A string passing over each

pulley supports a weight holder upon which weights may be placed. A pin holds a small ring to

which the strings are attached and which act as the particle. When a test for equilibrium is to be

made, the pin is removed; if the forces are in equilibrium the particles will not be displaced.

Theory:

A scalar is a physical quantity that possesses magnitude only: examples of scalar quantities are

length, mass and density. A vector is a quantity that possesses both magnitude and direction;

examples of vectors are velocity, acceleration and force. A vector

may be represented by drawing a straight line in the direction of the

vector, the length of the line being made proportional to the

magnitude of the vector; the sense of the vector, for example,

whether it is pointing toward the right or toward the left, is indicatedby an arrowhead placed at the end of the line.

Vectors may be added graphically. For example, if two or more

forces act at a point, a single force may act as the equivalent of the

combination of forces. The resultant is a single force which

produces the same effect as the sum of several forces, when these

pass through a common point (Fig. 2). The equilibrant is a force

equal and opposite to the resultant. A vector may also be broken up

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Fig. 3 - Graphical Method

Fig. 4 - Parallelogram Method

into components. The components of a vector are two vectors in different directions, usually at right

angles, which will give you the original vector when added together.

The operation of adding vectors graphically consists in constructing a figure in which a straight line

is drawn from some point as origin to represent the first vector, the length of the line being

proportional to the magnitude of the vector and the direction of theline being the same as the direction of the vector. From the

arrowhead end of this line and at the proper angle with respect to the

first vector, another line is drawn to represent the second vector and

so on with the remaining ones. The resultant is the vector drawn from

the origin of the first vector to the arrowhead of the last (Fig. 3). If a

closed polygon is formed, that is, if the arrowhead of the last vector

falls upon the origin of the first, then the resultant is zero. If the

vectors represent forces, they are in equilibrium.

Vectors may also be added analytically by calculating the x and y components of each vector, getting

the algebraic sum of all the x components and the algebraic sum of all the y components, and thencomputing the magnitude and direction of the resulting vector by using the Pythagoras theorem and

the definition of tangent, respectively.

To find the resultant of two vectors by the parallelogram

method, the two vectors A and B to be added are laid off

graphically to scale and in the proper directions from a common

origin, so as to form two adjacent sides of a parallelogram (Fig

4). The parallelogram is then completed by drawing by

drawing the other two sides parallel respectively to the first two.

The diagonal R of the parallelogram drawn from the same

origin gives the resultant, both in magnitude and direction.

These methods may be used for the addition of any number of vectors, by first finding the resultant

of two vectors, then adding the third one to this resultant in the same way and continuing the process

with other vectors.

Procedure:

1. Set up a force table as in Fig. 1. Make sure the table is leveled before starting the experiment.

Mount a pulley at the 20 mark on the force table and suspend a total of 100 grams over it.

Mount a second pulley at the 120 mark and suspend a total of 200 grams over it. Draw a vector

diagram to scale, using a scale of 20 grams per centimeter and determine graphically the

direction and magnitude of the resultant by using the parallelogram method.

2. Check the result of procedure 1 by setting up the equilibrant on the force table. This will be a

force equal in magnitude to the resultant, but pulling in the opposite direction. Set up a third

pulley 180 from the calculated direction of the resultant and suspend weights over it equal to the

magnitude of the resultant. Cautiously remove the center pin to see of the ring remains in

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equilibrium. Before removing the pin make sure that all the strings are pointing exactly at the

center of the pin, otherwise the angles will not be correct. The reason for doing this is to

compare the theoretical and experimental results.

3. Mount the first two pulleys as in procedure 1, with the same weights as before. Mount a third

pulley on the 220 mark and suspend a total of 150 grams over it. Draw a vector diagram to scaleand determine graphically the direction and magnitude of the resultant. This may be done by

adding the third vector to the sum of the first two, which was obtained in procedure 1. Now set

up the equilibrant on the force table and test it as in procedure 2.

4. Clamp a pulley on the 30 mark on the force table and suspend a total of 200 grams over it. By

means of a vector diagram drawn to scale, find the magnitude of the components along the 0 and

the 90 directions. Set up these forces on the force table as they have been determined. These

two forces are equivalent to the original force. Now replace the initial force by an equal force

pulling in a direction 180 away from the original direction. Test the system for equilibrium.

Calculations:

1. Calculate the resultant in procedure 1 by solving for the third side of a triangle algebraically. The

magnitude of the vector may be obtained by using the law of cosines, and the direction may be

obtained by using the law of sines.

2. Calculate the resultant in procedure 1 by using the analytical method of adding vectors.

3. Estimate errors of measurements by comparing:

a. Experimental values for the resultant forces to those obtained from graphical method.

b. Experimental values for the resultant forces to those obtained from analytical method.

Questions:

1. State how this experiment has demonstrated the vector addition of forces.

2. In procedure 3 could all four pulleys be placed on the same quadrant or in two adjacent quadrants

and still be in equilibrium? Explain.

3. State the condition for the equilibrium of a particle.

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MOTION OF A BODY IN FREE FALL

Apparatus:

SBehr Free-Fall apparatus

S Pre-made tape from the free fall apparatus

S Masking Tape

S Ruler and/or meter stick

Discussion:

In the case of free falling objects the acceleration and the velocity are in the same direction so that

in this experiment we will be able to measure the acceleration by concerning ourselves only with

changes in the speed of the falling bodies. (We recall the definition of acceleration as a change in

the velocity per unit-time and the definition of velocity as the displacement in a specified direction

per unit-time.)

A body is said to be in free fall when the only force that acts upon it is gravity. The condition of free

fall is difficult to achieve in the laboratory because of the retarding frictional force produced by air

resistance; to be more accurate we should perform the experiment in a vacuum. Since, however, the

force exerted by air resistance on a dense, compact object is small compared to the force of gravity,

we will neglect it in this experiment.

The force exerted by gravity may be considered to be constant as long as we stay near the surface

of the earth; i.e., the force acting on a body is independent of the position of the body. The force of

gravity (also known as the weight of the body) is given by the equation:

where mis the mass andgis the acceleration due to gravity

The direction ofgis toward the center of the Earth. As shown by Galileo, the acceleration imparted

to a body by gravity is independent of the mass of the body so that all bodies fall equally fast (in a

vacuum). The acceleration is also independent of the shape of the body (again neglecting airresistance).

Useful Information for Constant Acceleration:

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Fig. 1 - Behr Free-Fall Apparatus

This is the Behr Free-Fall apparatus. Initially the body is at the top of the post and held by an

electromagnet; when the switch is opened, the magnet releases the body, which then falls. A record

is made of the body's position at fixed time intervals by means of a spark apparatus and waxed tape.

When the body is released, it falls between two copper wires that are connected to the spark source.

The device causes a voltage to be built up periodically between the wires, and this causes a spark to

leap first from the high voltage wire to the body and to the ground wire, a mark is burned on the

waxed paper by the spark. The time interval between sparks is fixed (here it is 1/60 of a second); thus

the time interval between marks on the tape is also fixed and the marks on the tape record the

position of the body at the end of these intervals. See Fig. 2.

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Fig. 2 - Sample tape and demonstration of falling bob

When you obtain a tape, inspect it and draw a small circle around each mark made by the spark

apparatus to help with the identification of the position marks. You will obtain the acceleration of

gravity, g, by three methods. The difference in the methods is in the analysis of the data on the tape.

Method I:

1. Choose a starting point and from that point on, label your points, 1, 2, 3 . . . n.

2. Obtain the distance, S in cm between two successive points.

For example, assume the distance between points 3 and 4 is: 4.52cm.

Thus, S=4.52cm.

3. Obtain the average velocity over each of these distances.

Note that the time interval, t, between two successive points is 1/60 [s].

4. Obtain the successive changes in average velocities,V, then use these changes to compute the

acceleration for each particular change.

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5. Tabulate your data as follows:

n S

[cm]

t= n t

[s]

[cm/s]

V

[cm/s] [cm/s2]

1 1/60

2 2/60

3 3/60

.

.

.

.

.

.

n n/60

Note:

6. Obtain gby taking the average of the values of aon the 6thcolumn of your table. Stateg and the

% differenceof your result.

Method II:

1. Plot a graph of velocity, V, versus time, t; the independent variable should be plotted on theabscissa and the dependent variable along the ordinate.

2. Use a transparent straight-edge and draw a straight line that in your judgement best represents

the direction taken by the plotted points.

3. Determine the slope of this line. It should have the dimensions of cm/s2.

4. Convert your value of the slope to equivalent number of m/s2.

5. Estimate errors of g and compare your results with the theoretical value of g = 9.80 m/s2.

Method III:

1. From the tape, measure the distance between points 1 and 11 and divide by the time interval of

1/6 (why 1/6?). This will yield the instantaneous speed of the falling body at some intermediate

point (which one?).

2. Repeat for points 2 and 12, then 3 and 13, etc.

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3. Make a plot of the resulting velocity and the time t. Determine gfrom the slope of the graph,

compute the % difference.

Questions:

1. What are the advantages (or disadvantages) in Method 3 over that Method 2?

2. What are the advantages and analysis by Method 2 as compared with Method 1?

3. Does any part of the experiment show that all bodies fall with constant acceleration?

4. What is the significance of the constants in the equation relating v and t you plotted?

5. Why doesn't the graph of V versus t (Methods 2 and 3) go through the origin?

At what time did the body start to Fall?

With what velocity? Can you determine its position when it started?

6. What physical significance does negative time have in your equation relating V and t?

7. What would be the effect on your graphs of a change in the time interval between sparks?

What would be the effect if we had used a body with twice the original mass of the body to do

the experiment?

8. Of the three methods given which is the best? Support your answer.

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Fig. 1 - Set-up

HORIZONTAL PROJECTILE MOTION

Apparatus:

SOne long steel pole

S One short steel pole

S One table clamp

S One small V-groove clamp

S One right angle clamp

S Launching track

S Small steel ball

S Level and plumb bob

S Carbon paper

S 11" x 17" white paper and masking tape

S Meter stick and ruler

Introduction:

For a projectile near the surface of the earth the position of a particle in a trajectory is broken up into

itsXand Ycoordinates in the plane of the trajectory. Thus, we examine the most general vector

equation for displacement.

Eq. 1

from which we deduce two equations for the "X"displacement and the "Y"displacement.

Eq. 2

Eq. 3

In the case of a projectile fired horizontally (e.g., ball rolls off a table) there is no initial velocity in

the Y-direction. Hence, Voy= 0in above equations and we are left with

Eq. 4

Eq. 5

These are the position equations applicable to horizontal motion. They give the "X"distance and

the "Y"distance from a starting point at time "t. You are now to determine what the initial

horizontal velocity Vo in Eq.4 of a ball rolling off a launching track by making measurements of the

"X" and "Y" displacements and then studying various aspects of its trajectory.

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Fig. 2 - Horizontal Projectile

Procedure:

1. Let a ball roll off the launching track (in a high position) from a known position on the incline

and fall on a sheet of carbon paper placed atop a piece of plain paper. Measure the total distance

displaced in theXdirection from X=0 (use a plumb line at the point of launch to findX=0).

Measure the total Ydisplacement; the distance from the launching point to the table. Since thetime to cover the totalXand Ydisplacements is the same, useEq.(4)andEq.(5)to calculate Vo,

the initial horizontal velocity with your measured values.

2. Repeat the above procedure identically nine (9) more times and obtain an average value for the

horizontal velocity as well as an error from the average making sure you always start the roll

from the same point on the incline.

3. Obtain seven otherXand Ypoints of the trajectory by lowering the launching track. Take an

average of three rolls to determineXfor each value of Y. Plot all eight (8) points on a graph of

Xversus Y. This should show the trajectory of the ball after it leaves the track.

4. Use equations 4 and 5 to eliminate the variable t to obtain equation y=f(x). This is the

mathematical model of the trajectory. Plot it on the same graph of Procedure 3 provided you use

the value of V0you obtained earlier.

5. Estimate errors of the trajectory.

Questions:

1. Which graph is more precise?

2. How long was the ball in the air from the highest position of the launch? (Use Eq. 4 and 5 and

your data).

3. If you change the initial velocity do you expect the trajectory to change? (Use the equation to

prove this).

4. Even if you roll the ball from the same spot on the incline you get slightly different initial

velocities. Why?

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Fig. 1 - Set-up

Fig. 2 - Equilibrium of a Rigid Body

EQUILIBRIUM OF A RIGID BODY

Apparatus:

SMeter stick

S Three meter stick clamps

S Two 50gr hangers

S Slotted weights

S Pendulum clamp (black)

S Steel rod

S Table clamp

S Electronic balance

Introduction:

If a rigid body is in equilibrium, then the vector sum of the external forces acting on the body yields

a zero resultant and the sum of the torques of the external forces about any arbitrary axis is also equal

to zero. Stated in equation form:

Fx= 0 Fy= 0 = 0

In this experiment a meter stick is used as a rigid body to illustrate the application of the equations

of equilibrium.

Procedure:

1. Determine the weight of the meter stick by weighing it on the balance. Suspend the meter stick

in a horizontal position by means of two loads (m1and m2) as shown in Fig. 1.

2. Take all data necessary to determine the location of the c.g. of the bar by means of the principle

of moments.

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3. Determine the location of the c.g. of the bar.

4. Determine position of pivot point x with respect to center of mass (be aware of the sign of x).

5. Calculate position of pivot point from the condition of equilibrium.

6. Compare experimental result obtained in Procedure 4 with the theoretical prediction found in

Procedure 5.

7. Estimate errors.

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Fig. 1 - Set-up for block on a horizontal plane

Fig. 2 - Forces acting on the system

FRICTION

Apparatus:

SFriction block

S Friction board

S Pulley

S 50gr hanger

S Slotted weights

S String


S Meter stick or pend. protractor

Theory:

For a large class of surfaces the ratio of the (static and kinetic) frictional force, f, to the normal force,

N, is approximately constant over a wide range of forces. This ratio defines, for specific surfaces,

the coefficient of friction, namely:

In the static case when our applied force reaches a value such that the object instantaneously starts

to move we obtain the maximum frictional force or limiting value of the frictional force fmax.

We can now obtain the coefficient of static

friction defined as:

When the object is moving it experiences a

frictional force, fK which is less than the

static. Frictional experiments tell us that we

can (analogous to the static case) define a

coefficient of kinetic friction given by:

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Fig. 3 -Forces acting on a block on an inclined plane

Procedure:

1. Set up the equipment as in Fig. 1.

2. Weigh the block, W1.

3. Increase the weight W2until the block is on the point of sliding. Record the value of W2.

4. Repeat for five other values of W1by slowly increasing the blocks weight by adding masses to

its top.

5. Newton's Laws tell us:

a) N = W1 b) At point of sliding fmax= W2

6. Plot f maxversus N. Determine sfrom the slope of your graph.

7. Repeat procedures 1-6 above but this time adjust W2so that the block W1moves with constant

speed after it has been given an initial push. Plot the data and obtain kfrom your graph.

8. Set the block on an inclined plane and increase the angle of the plane until the block is on the

verge of sliding down. Note the value of the inclined plane angle.

9. Repeat Procedure 8 except with the inclined plane angle adjusted, Fig. 3, so that the block moves

down the plane with constant speed after it has been given an initial push. Note the value of the

angle at which this happens.

10. The data of Procedures 8 and 9 permit us to determine sand kby analyzing the forces on theblock in Fig. 3.

By applying Newtons Second Law we see that:

From which:

Use this last formula which permits us to obtain sfrom Procedure 8 and kfrom Procedure 9.

Estimate errors and compare the % difference of these with the values you obtained in 6 and 7.

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Questions:

1. Which coefficient, s, or kis usually the larger?

2. What graphical curve should you obtain in part 6?

3. Is it possible to have a coefficient of friction greater than 1? Justify your answer.

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Fig. 1 - Set-up for accelerating block on a horizontal plane

Fig. 2 - Forces acting on the system

NEWTON'S SECOND LAW

Apparatus:

SFriction board

S Friction block/box

S Pulley

S 50gr hanger

S Slotted weights

S String

S Long rod

S Small rod

S Table clamp

S Right angle clamp

S Stopwatch

SElectronic balance

S Meter stick

S Pend. protractor

Introduction:

Newton's Second Law can be written in vector form as

whereF is the vector sum of the external forces acting on a body and ais the resultant acceleration

of the c.g. of the body. IfF is constant, then ais constant and the equations of motion with constant

acceleration apply, i.e.,

With N=Normal force, T=Tension in the cord and

fk=Kinetic frictional force=kN then

F on System:F = m2g - fk= (m1+ m2) a

F= N - m1g = 0

F on m1:T - fk= m1a

F on m2:T - m2g = - m2a

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Fig. 3 - Forces acting on the system - with block m1accelerating up the inclined plane

Fig. 4 - Forces acting on the system with block m1accelerating down the inclined plane.

Therefore,

In the case of a block sliding down the plane we have:

The perpendicular forces acting on the system whether the block is accelerating up or down an

inclined plane are:

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Procedure:

1. Set up the equipment as in Fig. 1.

2. Compute the acceleration of m2by noting the time it takes for it to fall a measured distance.

From the acceleration and the known masses of m1and m2, compute the tension in the cord.Now apply Newton's Second Law to find the kinetic frictional force on m1and determine k.

Note: You may use a Photogate timer coupled with an Accessory Photogate timer set to pulse

mode to measure the time it takes for m1to travel down the plane for a given distance.

3. Repeat for four additional values of m2.

4. Set up the apparatus as indicated in Fig. 5. Arrange m1and m2so that m1accelerates down the

plane. Note m1, m2and . Measure the acceleration of the system. Calculate the accelerationof the system. (Use kobtained from fkfound in Procedure 2) and compare with your measured

value. Express % difference and % error.

5. Repeat Procedure 4 for a different value of and adjust m1and m2 so that m1accelerates up theplane. Remember to write a new force equation for an object accelerating up the plane.

Fig. 5 - Set-up for m1accelerating down an inclined system

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Fig. 2 - Simple Pendulum

THE SIMPLE PENDULUM

Apparatus:

STable clamp

S Steel rod

S Pendulum clamp (silver)

S Pendulum bob (various sizes)

S String


S Master photogate timer (set to pendulum mode)

S Meter stick

S Pend. Protractor

Introduction:

A simple pendulum consists of a small mass (the pendulum bob) suspended by a non-stretching,

massless string of length L. The period Tof oscillation is the time for the pendulum bob to gofrom one extreme position to the other and back again.

Consider the variables that determine the

period of oscillation of a pendulum:

S The amplitude of oscillation. The

amplitude of the pendulums swing is the

angle between the pendulum in its

vertical position and either of the

extremes of its swings.

S The length L of the pendulum. The

length is the distance from the point of

the suspension to the center (of mass) of

the pendulum bob.

S The mass m of the bob.

S The acceleration due to gravity g.

From unit analysis we can show:

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Where T = period of oscillation;

m = mass of bob;

L = length of string;

g = acceleration due to gravity

Since an oscillation is described mathematically by cos t and knowing that =2f where

we then have:

[Eq. 1] can be re-written as: [Eq. 2]

Procedure:

Make the following measurements:

1. Turn on the photogate and set it to pendulum mode. In addition, make sure the memory switchon as well. Set-up the pendulum so that when it is in resting position it blocks the photogate as

shown on Fig. 1.

2. Period as a function of amplitude (plot T vs. ). Perform this procedure for amplitudes of 5o

to 30oin steps of 5o. At each given angle allow the pendulum to swing through the photogate,

be careful not to strike the photogate with the pendulum. For each amplitude record the period

as displayed on the photogate. The length and mass will remain constant.

3. Period as a function of length (plot T vs. L). Use a small amplitude such as 10o. Each time

a new length, L is set, the length must be measured from the center of the bob to the pivot point.

For each length record the period as displayed on the photogate. The amplitude and the mass willremain constant. Fit the data to Eq. 2. How does the obtained g value from the fit compare

to the known value of g?

4. Period as a function of mass (plot T vs. m). Use a small amplitude such as 10o. Use 4

different masses but keep amplitude and length constant. For each mass record the period as

displayed on the photogate.

Questions:

1. For the simple pendulum where is the maximum for: displacement, velocity and acceleration?

2. Would the period increase or decrease if the experiment were held on :

a) top of a high mountain?

b) the moon?

c) Jupiter?

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Fig. 1 - Centripetal Force Apparatus and Display of Static Test

CENTRIPETAL FORCE

Apparatus:

SCentripetal force apparatus

S Set of slotted weights

S 50g hanger

S Stopwatch


S Level

S Ruler

Theory:

A mass m moving with constant speed v in a circular path of radius r must have acting on it a

centripetal force F where n is the revolutions per sec.

Eq. 1

Description:

As indicated in Fig. 1, the shaft, cross arm, counterweight, bob, and spring are rotated as a unit. The

shaft is rotated manually by twirling it repeatedly between your finger at its lower end, where it is

knurled. With a little practice it is possible to maintain the distance r essentially constant, as

evidenced for each revolution by the point of the bob passing directly over the indicator rod. The

centripetal force is provided by the spring.

The indicator rod is positioned in the following manner: with the bob at rest with the spring

removed, and with the cross arm in the appropriate direction, the indicator rod is positioned andclamped by means of thumbscrews such that the tip of the bob is directly above it, leaving a gap of

between 2 and 3mm.

The force exerted by the (stretched) spring on the bob when the bob is in its proper orbit is

determined by a static test, as indicated in Fig. 1(Static Test).

The mass m in Eq. 1 is the mass of the bob. A 100-gm mass (slotted) may be clamped atop the bob

to increase its mass. The entire apparatus should be leveled so that the shaft is vertical.

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Fig. 2 - Centripetal Force Apparatus Rotating

Procedure:

Devise a method for determining whether the shaft is vertical, and make any necessary adjustments

of the three leveling screws.

The detailed procedure for checking Eq. 1experimentally will be left to the student.

At least three values of r should be used,

with two values of m for each r. A

method for measuring r should be thought

out, for which purpose the vernier caliper

may be useful. The value of n should be

determined by timing 50 revolutions of

the bob, and then repeating the timing for

an additional 50 revolutions. If the times

for 50 revolutions disagree by more than

one-half second, either a blunder incounting revolutions has been made, or

the point of the bob has not been

maintained consistently in its proper

circular path.

Results and Questions:

1. Exactly from where to where is r measured? Describe how you measured r.

2. Tabulate your experimental results.

3. Tabulate your calculated results for n, F from static tests, and F from Eq. 1, and the % difference

between the F's, using the static F as standard. Estimate % error.

4. Describe how to test whether the shaft is vertical without the use of a level. Why should it be

exactly vertical?

5. Why is the mass of the spring not included in m?

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WORK AND KINETIC ENERGY

Apparatus:

SLinear air-track and air-track kit

S Two photogate timers (Master)

S Variable air supply and hose

S Two gliders


Theory:

Work and kinetic energy are related as:

If the total work done by all the forces on the system is zero, the total kinetic energy remainsconstant. Consider a system of two masses, m1and m2moving at velocities, v1iand v2i, about to

undergo a collision. Before the collision the kinetic energy is:

After the collision the two bodies move off with velocities v1fand v2frespectively. Thus, the kinetic

energy after the collision is:

Pre-lab Exercise: Before the day of the experiment run the simulation below to get a better

understanding of how the experiment will work. Perform the simulation for the conditions as listed

on page 48 (note, if you want a cart to move to the left, you must make its velocity negative):

http://www.mrwaynesclass.com/teacher/Impulse/SimFriction/home.html

Procedure:

1. Set-up the air track and photogates as shown in Fig. 1. Carefully level the track.

Fig. 1 - Air-Track Set-up

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2. Measure m1and m2, the masses of the two gliders (include the flags and bumpers) to be used in

the collision. Record your results in tabular form. Start with equal masses then use attachments

to change the masses as required in Procedure 6 below.

3. Record the length of the flags on the gliders as 0.1m each.

4. Set both Photogate Timers to GATE mode, and press RESET buttons. Make sure the memory

switch is on.

5. Estimate the % error in time by letting a glider go through the two photogates. Calculate the %

error in the time measurement (If an older model of photogate is used set it to 0.1ms resolution,

make sure you switch it back to 1ms resolution before starting Procedure 6).

6. The experiment will be performed for the following configurations:

1. m1= m2 v1i0 v2i= 0

2. m1> m2 v1i0 v2i= 0

3. m1< m2 v1i0 v2i= 04. m1> m2 v1i= 0 v2i= 0 ( explosion )

5. m1> m2 v1i0 v2i0

6. m1> m2 v1i0 v2i= 0 ( coupled )

Note, when increasing masses on a glider it must be done symmetrically.

For each configuration record four time measurements as follows:

t1i = the time that glider1blocks photogate1before the collision.


(In cases where v2i= 0 there is no t2isince glider2begins at rest.)

t1f = the time that glider1blocks photogate1after the collision.t2f = the time that glider2blocks photogate2after the collision.

IMPORTANT:The collision must occur after glider1has passed through photogate1and, after the

collision, the gliders must be fully separated before either glider interrupts a photogate.

NOTE: Use the memory function in the ME-9215 Photogate Timer, to store the initial times while

the final times are being measured. The numbers on the screen reflect the first time the glider

crossed the photogate. When you flip the memory switch you get a second reading which represents

the total time a glider traveled through a photogate forth and back (after collision). Subtract the first

number from the second to obtain the final time, tf.

Calculations and Questions:

For each case:

1. Calculate the Initial Kinetic Energy, KEi, and the error in KEi.

Percent error in the Kinetic Energy = (2)*( % error in time)

2. Calculate Final Kinetic Energy, KEfand the error in Kef.

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CONSERVATION OF MOMENTUM

Apparatus:

SLinear air-track and air-track kit

S Two master photogate timers

S Two gliders

S Variable output air supply and hose


Theory:

When objects collide, whether locomotives, shopping carts, or your foot and the sidewalk, the results

can be complicated. Yet even in the most chaotic of collisions, as long as there are no external

forces acting on the colliding objects, one principle always holds and gives up an excellent tool for

understanding the dynamics of the collision. That principle is called the conservation of momentum.

For a two-object collision, momentum conservation is easily stated mathematically by the equation:

where m1and m2are the masses of the two objects, v1iand v2i are the initial velocities of the objects

(before the collision), v1fand v2fare the final velocities of the objects, and piand pfare the combined

momentums of the objects, before and after the collision. In this experiment, you will verify the

conservation of momentum in a collision of two air track gliders.

Pre-lab Exercise: Before the day of the experiment run the simulation below to get a better

understanding of how the experiment will work. Perform the simulation for the conditions as listed

on page 50:

http://www.mrwaynesclass.com/teacher/Impulse/SimFriction/home.html

Procedure:

1. Setup the air track and photogates as shown in Fig.1, using bumpers on the gliders to provide an

elastic collision. Carefully level the track.

2. Record the length of the flags on the gliders as 0.1m each.

Fig. 1 - Air-Track Set-up

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3. Set both Photogate Timers to GATE mode, and press RESET buttons.

4. The experiment will be performed for the following configurations where each glider will have

a rubber band bumper except for configuration 4.5:

4.1. m1= m2 v1i0 v2i= 0

4.2. m1> m2 v1i0 v2i= 04.3. m1< m2 v1i0 v2i= 0

4.4. m1> m2 v1i0 v2i0

4.5. m1> m2 v1i0 v2i= 0 ( coupled )

The last configuration is an example of non-elastic collision in which two gliders, instead of

bouncing off each other, stick together so that they move off with identical final velocities.

Replace the rubber bumpers with the wax and needle attachments.

5. Measure m1and m2, the masses of the two gliders (include the flags and bumpers) to be used in

the collision. Record your results in tabular form.

Note, when increasing masses on a glider it must be done symmetrically.

6. Four time measurements will be obtained for each configuration



(In cases where v2i= 0 there is no t2isince glider2begins at rest.)

t1f = the time that glider1blocks photogate1after the collision.

t2f = the time that glider2blocks photogate2after the collision.

IMPORTANT:The collision must occur after glider1has passed through photogate1and, after the

collision, the gliders must be fully separated before either glider interrupts a photogate. Use the

memory function to store the initial times while the final times are being measured.

Data and Calculations:

11.. For each time that you measured, calculate the corresponding glider velocity (v=L/t) where the

velocity is positive when the glider moves to the right and negative when it moves to the left.

2. Use your measured values to calculate piand pf, the combined momentum of the gliders before

and after the collision.

3. Estimate error of piand pfand conclude if pi=pfwithin these errors.

Questions:

1. Was momentum conserved in each of your collisions? If not, try to explain any discrepancies.

2. Suppose the air track was tilted during the experiment. Would momentum be conserved in the

collision? Why or why not?

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Fig. 1 - Set-up

ROTATIONAL MOTION AND MOMENT OF INERTIA

Apparatus:

- Friction board and block

- Solid cylinders- Hollow cylinders

- Solid spheres

- Hollow spheres (e.g. pin-pong ball)

- Photogates (master and accessory) timers

- Protractor (from Pendulum exp.)

- Two meter sticks

Theory:

The motion of a rigid body is the combination of translation and rotation. The rotational inertia, I,

of a body rotating about a fixed axis measures the resistance of the body to angular accelerationwhen a torque is applied. The torque, , is the product of force times lever arm (perpendicular

distance from the line of action of the force to the axis of rotation). The angular acceleration, , is

the rate of change of angular velocity. The variation of the moment of inertia as a function of mass

distribution or shape of the object will be investigated.

Rotational Inertia, I, about the axis of symmetry

Solid Cylinder Hollow Cylinder Solid Sphere Hollow Sphere

*The moment of inertia is to rotational motion what the mass of an object is to translational motion.

This analogy can be illustrated as follows:

Translational Motion

F = ma Analogy

Rotational Motion

= I

F driving force

m resistance to changes in velocities I

a acceleration or response to changes in velocities

Consider an object rolling down an incline plane having an angle (Fig. 2) the following equations

apply:

- The sum of the forces yielding the objects translational acceleration aalong the ramp is given

by:

F = mg sin -f= ma Eq. 1

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- The sum of the torques providing the

the objects acceleration about its

center can be written:

=fR = I Eq. 2

- Because the objects roll without

slipping, one also has the following

relationship between the translational

and rotational accelerations:

a = R Eq. 3

take:

I = k mR2 and ; object rolled down incline of length, s, with v0= 0

Eq. 4

Procedure:

1. Set up the inclined plane as in Fig. 1 using an angle of about 5o to 6o. Use the photogates in a

coupled mode, set the master photogate to pulse mode.

2. Measure the distance between the photogates and the angle.

3. Measure time t at least five different times, for the different objects provided by noting the time

it takes to roll between the two photogates starting from rest.

Calculations and Excersises:

For each object:

1. From procedure 3 using the measured times calculate the corresponding k value for each object

using Eq. 4.

2. Evaluate the % error in k.

3. Find % differences in k for all shapes studied.

4. Make the conclusion about how close (or how far) your measured k values are to (from) the

given ones as listed on Table 1.

*Adapted from PHY 211 - Lab 6 - Univeristy of Illinois up to Eq. 3

Fig. 2

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SUPPLEMENTARY

EXPERIMENTS

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Eq. 1

Eq. 2

DENSITY AND ARCHIMEDES' PRINCIPLE

Apparatus:


S Hook stand and beaker base

S 600ml plastic beaker

S Aluminum cylinder with hook

S Brass pendulum bob (2.54cm)

S String

S 500ml graduated cylinder with water

S Ruler or Vernier caliper

Introduction:Archimedes's principle states that a body immersed in a fluid is buoyed up by a

force equal to the weight of the displaced fluid (Fig. 2).

Imagine now that a body is suspended in water as shown in Fig. 3b. The

effective weight of the body (Weff) as measured by the masses on the scale is

given by:

but the effective weight is equal to the weight of the object minus the buoyant

force.

Fig. 1 - Set-up

Fig. 2

Fig. 3 - (a) Object in air (b) Object submerged in water

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Eq. 3

Eq. 4

Eq. 5

Eq. 6 Eq. 7

By Archimedes's principle:

Substituting Eq. 3 into Eq. 2:

If the cross sectional area is constant, Eq. 7 reduces to:

Eq. 8

M' can be varied by varying Li(length of cylinder submerged in water) and the data can thus be

obtained so that can be determined graphically.

Procedure:

1. Measure the diameter and length of the aluminum cylinder (Fig. 4a). Obtain the mass of the

cylinder (Fig. 4b). Calculate the density of your object by direct measurement of mass and

volume, compare to the known density.

2. Carefully add water to the beaker varying Liin steps of 1cm and recording the corresponding M

each time (Fig. 4c). Do this from 1cm up to 7cm. Plot a graph of M' vs. Li (Eq. 8).

Determine the density of the object from the slope of your line and compare to the known

density.

Fig. 4 - (a) Obtaining cylinder dimensions, (b) Obtaining mass, M of cylinder, (c) Obtaining mass M

of cylinder corresponding to Li

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3. If the body is completely submerged in water, Vi= V, Eq. 7 reduces to:

Eq. 9

Use a brass ball as your object. Weigh the ball (Fig. 5a). Carefully add water to the beaker till

the ball is completely submerged (Fig. 5b). Be sure the ball is not touching the bottom of the

beaker. Record the mass M' in grams. Using Eq. 9 determine the density of the brass ball and

compare it to the known value.

Questions:

1. How do the two values of density from Procedures 1 and 2 compare with each other?

2. Derive Eq. 9.

3. How do the errors in procedure 1 and 2 compare to the error in procedure 3?

4. Archimedes is supposed to have discovered the principle which bears his name when he was

asked whether a certain crown was made of gold. How could you solve Archimedes' problem

using an unmarked balance (i.e. you cannot read weight; you can only balance two objects

against each other). Use Archimedes' principle in your solution.

Fig. 5 - Applying Archimedes Principle to a Brass ball

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Fig. 1 - Set-up

Fig. 2 - Vector diagrams due to the collision

COLLISION IN TWO DIMENSIONS

Apparatus: (Set-up is similar to that of Horizontal Projectile Motion)

SOne long steel rod

S One small rod

S One small V-groove clamp

S One right angle clamp

S Launching track with mounting screw

S Steel and glass spheres

S Plumb bob, level

S 11" x 17" paper, carbon paper

S Masking tape

S Meter stick

S 18" ruler

SElectronic balance

Introduction:.

A steel ball traveling with velocity Vostrikes a stationary ball. After impact the two balls depart with

velocities V1' and V2'.

Conservation of Momentum in the X and Y direction:

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Fig. 3 - Sequence of Events: Prior, During and After Collision

Procedure:

1. Determine Vo: Let the steel ball roll off an inclined track a measured height, h, from the top of

the table. The line constructed from a point directly below the edge of the track to the point of

impact determines the X - axis. The distance from the point directly below the edge of the track

to the point of impact is used to calculate the velocity Vo.

2. Place a ball on the mounting screw at the same height so that it is level with the ball coming

down the track at collision. Be careful that collision takes place without either ball hitting the

platform.

Measure the X and Y distance traveled for both balls and calculate V1x', V1y', V2x', V2y'.

3. Verify conservation of momentum in both the X and Y direction.

4. Repeat for a steel ball striking a marble.

5. Explain the sources of error and estimate the errors in the experiment.

6. Explain why it is important that the center of mass of the two balls has to be at the same vertical

height.

7. Calculate the initial and final kinetic energies for each collision. Express % difference.

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Phy 116 Lab Manual Fall 15

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Transcript of Phy 116 Lab Manual Fall 15