Lab VII

Potential and Kinetic Energy

1 Introduction

This is a lab about the interplay between kinetic and potential energy. While we cancalculate forces and accelerations of an object as it moves along a complicated path,the use of energy gives us a way to bypass these complex calculations, and understandmotion in a simpler way.

1.1 Forms of Energy

The kinetic energy T of an object can be written as:

T =1

2mv2 (VI.1)

where m is the mass of the object and v is its velocity.The potential energy that we will be dealing with is gravitational kinetic energy

Ug, and can be written as:Ug = mgh (VI.2)

where, again, m is the mass of the object, g is the acceleration of gravity (approxi-mately 9.8 m/s2 on earth) and h is the object’s height above some defined position.We can select where h = 0 for our own convenience, because only changes in potentialenergy are significant.

The total energy of our object is then given by:

E = T + Ug. (VI.3)

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1.2 Energy conservation

If our total energy is conserved, then E doesn’t change and we can write ∆E = 0.This implies that:

∆T = −∆Ug (VI.4)

So in the motion between points ‘1’ and ‘2’:

T2 − T1 = −(Ug2 − Ug1) = mg (h1 − h2) (VI.5)

or1

2m

(

v2

2 − v2

1

)

= mg (h1 − h2) . (VI.6)

If an object starts from rest, v1 = 0 and T1 = 0. So:

v2

2 = (2g)h1 − (2gh2) (VI.7)

and if we plot v22

vs. h1, we should get a slope of 2g and an intercept of −2gh2.

1.3 Energy losses

Among the ways that an object can lose energy, friction and drag are the mostimportant. In both cases, the object does work W :

W = F · d (VI.8)

when it exerts a force F over distance d. While we will be working to minimize theeffects of friction and drag on our measurements, they will always be present to somedegree.

The force from kinetic friction is approximately constant for a given object, so thework done (and energy lost) by friction will therefore be proportional to the distancetraveled. The force of friction Ff is related to the ‘normal’ force Fn (the force of thecar on the track) by

Ff = µkFn (VI.9)

where µk is the kinetic coefficient of friction. The normal force comes (in part) fromthe weight of the car, but a car traveling on a track that curves upward (for example)has additional normal force coming from the car being accelerated upward as it followsthe curve.

Drag forces are proportional to v2 (or v, for small or slow objects); to see the effectof drag forces will be more complicated.

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We can now update our expression for the energy of an object:

E = T + Ug − Wf (VI.10)

where Wf is the work done by the object in the form of friction or drag.Note that while total energy is conserved, friction and drag are energy lost by our

object because the work done has generated heat or other forms of energy.

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3 Equipment

3.1 Track

The track may already be set up before you start. Should you need to set up orchange the track configuration, please follow these guidelines.

There are three boards for holding track, assemble them with the smaller boardin the middle, with the plastic support bases located at the joint between the boards.You may also have binder clips to hold the board together at the top. There arecolored dots on the upper corners of the boards: match the dots on the corners thatgo together.

The track is supported by plastic pegs, which come in two varieties: ‘photogate’pegs, which have a screw fitting for holding a photogate, and ‘normal’ pegs that haveplain end.

Figure VII.1: Track support pegs; photogate support peg is on top.

You must first attach a peg-holder inside the hole in the board, before you caninsert a peg. The peg-holder is a round insert with a ‘nut’ that tightens on the backof the board. Please note: there is a groove inside the peg-holder that fits a pin thatprotrudes from the ‘photogate’ pegs to keep them from twisting. Insert the photogatepegs carefully so that they have their pin in the groove.

The track is attached to the pegs by clips. You put the clips in the bottom of thetrack by turning them so that the flat tab part of the clip attaches to the grooves onthe bottom of the track. Then attach the other end of clip to the peg. You may needto slide the clip along the track slightly to get it in the right position.

One of your lab group should be stationed at the end of the track as a ‘catcher’to stop the car before it falls off the end of the track. The catcher should get theirhands in place before the car is released!

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Figure VI.2: Track clip.

Figure VII.3: Car with photogate flag.

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Figure VII.4: Photogate for time and velocity measurement.

3.2 Cars

The cars are quite simple, with three wheels that fit in grooves on the top of thetrack. Make sure the wheels are in the right place, or the car will have extra frictionor fall off the track!

There are slots on the sides of the car for fitting a small vertical ‘flag’ (see Fig.VII.3). This flag is used with the photogates to measure the velocity of the cars. Put only one flag on a car.

Weights can be attached to the underside of the cars, using a screw that threadsinto the brass weight. Drag has less of an effect when the mass of the car is increased,so it’s a good idea to use the added weight.

3.3 Photogates

Photogates are attached to the special ‘photogate’ pegs and used for timing and velocity measurements (see Fig. VII.4). There are small holes (one of which is visible in the figure) for the an infrared beam that is blocked by the ‘flag’ on the cars.

Adjust the photogates so that the body of the photogate is perpendicular to thetrack, and the height of the infrared beam above the track is the right height so thatthe cars flag (and not the car body) blocks the beam. You can put the timer in ‘test’mode to check if the flags are correctly blocking the infrared beam.

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Figure VII.5: Timer module used for time and velocity measurement.

3.4 Timer

Plug the photogate into the timer (see Fig. VII.5), using the input labeled ‘1’ on the right side of the timer. Connect the timer’s power supply on the left side of the timer, and switch it on.

Using the control buttons on the timer, select ‘Velocity’, and ‘One photogate’.Before each measurement it is necessary to press the ‘START’ button on the timer.

The timer gives velocities in cm/s.

4 Procedure

4.1 Consistency check

First we must check the consistency of our measurements. Set up the track with adownward slope leading to a long level segment of track. This is track configurationwith the peg positions marked on the board with red dots (See Fig. VI.6). Put thephotogate peg and the photogate in the position marked with two red dots.

We want to start the car from rest at the same place on the track, so pick one ofthe pegs in the downward slope as your starting point. Attach a small binder clip tothe track to mark the starting position of the back of the car.

The main point is that a car released from the starting point will have a consistentvelocity along the level part of the track. You should make sure that when the caris at the starting point, the car is against the binder clip, and that when the car isreleased it is neither pushed nor held back.

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Figure VII.6: Track configuration #1, marked with red dots

Connect the photogate to input 1 of the timer and set the timer to measurevelocity.

Press START on the timer and release the car from the starting point. Recordthe velocity of the car in Table VI.1. NOTE: the timer gives you velocities in cm/s,convert to m/s when you enter the velocity measurements in the data tables.

Repeat the measurement two times, entering the table in Table VI.1. Calculate the average of your measurements and enter it in the Table.

4.2 Potential to Kinetic Energy 1

Keep the track configuration unchanged for this part.Weigh your car and enter the weight in Table VI.2. Then take the meter stick

and measure the height from the tabletop to the track at the photogate, and enterthe value as h2 in Table VI.2. NOTE: that part of the track may be on top of themotion table cover, rather than the lab bench. Measure the height of the cover, andadd to the height from the top of the cover to your starting position.

Select three starting points for your car, Your starting points should be spread out along the downward sloping part of the track to get a full range of heights.

Measure the approximate horizontal distance between your starting points (takethe middle starting point) and the photogate, and record it as the d in Table VI.2.

At each starting point measure the height of the track, then center the car abovethe starting point. Release the car and take a velocity measurement. Enter the height(h1) and velocity (v2) in Table VI.2.

Calculate the car’s kinetic energy from its velocity in Table VII.2. Note that the timer gives velocities in cm/s, and you must convert to m/s when calculating the kinetic energy. Calculate the change in potential energy ∆U between the starting point and the photogate and enter it in the table.

Make a graph of the car’s velocity squared (v22

on the y-axis) vs. h1 (x-axis) and

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Figure VI.7: Track configuration #2, marked with yellow dots.

fit with a straight line. Enter the slope and intercept in Table VI.2, calculate g fromyour fit, and get a value for h2 from the intercept of your graph (see Eqn. VI.7).

Attach your graph to the data you hand in.

4.3 Potential and Kinetic Energy 2

Now adjust your track configuration to the one marked with yellow dots. As before,put the photogate in the spot marked with two yellow dots (See Fig. VI.7).

Now repeat the previous series of measurements, entering your data in Table VI.3,and plotting and fitting the data the same way as you did previously. Try to get yourstarting points close to the heights used for your measurements using the first trackconfiguration.

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Name: Sec/Group: Date:

5 Data

Table VII.1: Consistency check

measurement # velocity (m/s)

1

2

average

Table VII.2: Potential and Kinetic Energy 1

h2: m mcar : kg d: m

h1 [m] v2 [m/s] T2 [J] ∆U [J]

Fit results:

slope: m/s2

intercept: m2/s2

g: m/s2

h′

2m

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Name: Sec/Group: Date:

Table VII.3: Potential and Kinetic Energy 2

h1 [m] v2 [m/s] T2 [J] ∆U [J]

Fit results:

slope: m/s2

intercept: m2/s2

g: m/s2

h′

2m

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Name: Sec/Group: Date:

6 Conclusions

1. The two track configurations differ in their shape; compare the speeds measuredfor the two tracks when the starting heights are close to the same. How do youexplain what you observed?

2. Calculate the potential energy difference for a car going between one startingpoint (pick one from your data) and the photogate, and compare to the finalkinetic energy you measured. Can you explain why there might be a differencebetween the energies?

3. How good a value for g do you get from your data? What systematic errorsmight be present? Suggest some techniques for reducing these errors.

4. If the frictional force on the car was 0.1 N, estimate how much energy wouldbe lost by the car during its travel. Now work out how much frictional forceis needed to account for any difference you measured between initial potentialenergy and final kinetic energy.