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Wilkes Science In Motion

Computer 4

Determining g on an Incline (Motion Detector)

During the early part of the seventeenth century, Galileo experimentally examined the concept of acceleration. One of his goals was to learn more about freely falling objects. Unfortunately, his timing devices were not precise enough to allow him to study free fall directly. Therefore, he decided to limit the acceleration by using fluids, inclined planes, and pendulums. In this experiment, you will see how the acceleration of a rolling ball or cart depends on the incline angle. Then, you will use your data to extrapolate to the acceleration on a vertical “incline;” that is, the acceleration of a cart dropped in free fall.

If the angle of an incline with the horizontal is small, a cart rolling down the incline moves slowly and can be easily timed. Using time and position data, it is possible to calculate the acceleration of the cart. When the angle of the incline is increased, the acceleration also increases. The acceleration is directly proportional to the sine of the incline angle, θ. A graph of acceleration versus sin(θ) can be extrapolated to a point where the value of sin(θ) is 1. When sin(θ) is 1, the angle of the incline is 90°. This is equivalent to free fall. The acceleration during free fall can then be determined from the graph.

Galileo was able to measure acceleration only for small angles. You will collect similar data. Can these data be used in extrapolation to determine a useful value of g, the acceleration of free fall? We will see how valid this extrapolation can be. Rather than measuring time, as Galileo did, you will use a Motion Detector to determine the acceleration. You will make quantitative measurements of the motion of a cart rolling down inclines of various small angles. From these measurements, you should be able to decide for yourself whether an extrapolation to large angles is valid.

Figure 1

OBJECTIVES Measure the velocity and acceleration of a cart rolling down an incline. Determine the mathematical relationship between the angle of an incline and the acceleration

of a cart rolling down the incline. Determine the value of free fall acceleration, g, by using an extrapolation on the acceleration

vs. sine of track angle graph. Determine if an extrapolation of the acceleration vs. sine of track angle is valid.

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MATERIALScomputerVernier computer interfaceLogger ProMotion DetectorVernier Dynamics TrackMotion Detector BracketAdjustable End StopVernier Dynamics Carthard ball, approximately 8 cm diameter rubber ball, similar sizemeter stickbooks

PRELIMINARY QUESTIONS1. One of the timing devices Galileo used was his pulse. Drop a rubber ball from a height of

about 2 m and try to determine how many pulse beats elapsed before it hits the ground. What was the timing problem that Galileo encountered?

2. Now measure the time it takes for the rubber ball to fall 2 m, using a watch or clock with a second hand or seconds display. Did the results improve substantially?

3. Roll the hard ball down an incline that makes an angle of about 10° with the horizontal. First use your pulse and then your watch or clock to measure the time of descent.

4. Do you think that during Galileo’s day it was possible to get useful data for any of these experiments? Why?

PROCEDURE1. Connect the motion detector to a digital (DIG) port of the interface. Set the motion detector

sensitivity switch to Cart.

Figure 2

2. Set up the equipment and place a single book under one end of the Dynamics Track so that it forms a small angle with the horizontal (see Figure 1). Adjust the points of contact of the two ends of the incline so that the distance, x, in Figure 1, is between 1 and 2 m.

3. Place the motion detector at the top of the incline. Position it so the Dynamics Cart will never be closer than 0.15 m.

4. Open the file “04 Determining g” from the Physics with Vernier folder.

5. Hold the cart on the incline about 0.25 m from the Motion Detector.2 Physics with Vernier


6. Click to start data collection; release the cart after the motion detector starts to click. Move your hand out of the motion detector path quickly. You may have to adjust the position and aim of the motion detector several times before you get it right. Adjust and repeat this step until you get a good run showing an approximately constant slope on the velocity vs. time graph during the rolling of the cart.

7. Fit a straight line to a portion of your data. First, select the portion by dragging across the graph to indicate the starting and ending times. Then, click Linear Fit, , to perform a linear regression of the selected data. Use this tool to determine the slope of the velocity vs. time graph using only the portion of the data for times when the cart was freely rolling. From the fitted line, find the acceleration of the cart. Record the value in your data table.

8. Repeat Steps 5–7 two more times.

9. Measure the length of the incline, x, which is the distance between the two contact points of the incline (see Figure 1). Record the length in your data table.

10. Measure the height, h, of the book(s). Record the height in your data table. These last two measurements will be used to determine the angle of the incline.

11. Raise the incline by placing a second book under the end. Adjust the books so that the distance, x, is the same as the previous reading.

12. Repeat Steps 5–10 for the new incline.

13. Repeat Steps 5–11 for 3, 4, and 5 books.

DATA TABLE

Number of books

Height of books, h

(m)

Length of incline, x

(m)sin(θ)

Acceleration Average acceleration

(m/s2)Trial 1(m/s2)

Trial 2(m/s2)

Trial 3(m/s2)

1

2

3

4

5

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ANALYSIS1. Using trigonometry and your values of x and h in the data table, calculate the sine of the

incline angle for each height. Note that x is the hypotenuse of a right triangle.

2. Calculate the average acceleration for each height. Plot a graph of the average acceleration (y-axis) vs. sin(θ). Use either Page 3 of the experiment file or graph paper. Carry the horizontal axis out to sin(θ) = 1 (one) to leave room for extrapolation.

3. Draw a best-fit line by hand or use the proportional fit feature of Logger Pro and determine the slope. The slope can be used to determine the acceleration of the cart on an incline of any angle.

4. On the graph, carry the fitted line out to sin(90°) = 1 on the horizontal axis and read the value of the acceleration.1

5. How well does the extrapolated value agree with the accepted value of free-fall acceleration (g = 9.8 m/s2)?

6. Discuss the validity of extrapolating the acceleration value to an angle of 90°.

EXTENSIONS1. Use the motion detector or video analysis to measure the actual free fall of a ball. Compare the

results of your extrapolation with the measurement for free fall.

2. Compare your results in this experiment with other measurements of g. For example, use the experiment, "Picket Fence Free Fall," in this book.

3. Investigate how the value of g varies around the world. For example, how does altitude affect the value of g? What other factors cause this acceleration to vary from place to place? How much can g vary at a school in the mountains compared to a school at sea level?

1Notice that extrapolating to the y value at the x = 1 point is equivalent to using the slope of the fitted line.4 Physics with Vernier


INSTRUCTOR INFORMATION 4 Experiment

Determining g on an InclineThere are two versions of this experiment included in the print book and in the electronic resources that accompany the book. One version uses the Motion Encoder and the other uses a Motion Detector. Use the version appropriate for your equipment.See Appendix A for information about the word-processing files of the student experiments, as well as any other electronic resources available for this book.

RELATED SKILLSSelect a region of data Add curve fits to data

ESTIMATED TIMEWe estimate that data collection and analysis for this experiment can be completed in one45-minute class period. If you are doing data analysis using Logger Pro, you may need additional time.

EQUIPMENT TIPS1. The Materials list includes a Vernier Dynamics Track and an Adjustable End Stop. The

Dynamics Cart and Track System, available from Vernier Software & Technology (order code: DTS or DTS-EC), includes both of these items. The DTS version is designed for use with the Motion Detector, while the DTS-EC version includes the Motion Encoder Cart and Receiver.

2. In 2016, Vernier Software & Technology stopped producing the Vernier Dynamics System (order code: VDS) with black aluminum tracks and green metal dynamics carts and started producing the Dynamics Cart and Track System (order code: DTS) with gray aluminum tracks and molded plastic dynamics carts.

3. The rubber ball and hard ball listed in the Materials section are used only in the Preliminary Questions. A lacrosse ball and a racquetball are two good options. If the velocity and acceleration graphs are noisy, try to increase the strength of the ultrasonic reflection from the target by increasing the target’s area and ensuring the target creates a strong reflection of the ultrasound.

If you begin with a velocity or acceleration graph and obtain a confusing display,switch back to the position graph for troubleshooting.



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Experiment 4

DATA COLLECTION AND ANALYSIS TIPS1. In the computer analysis, we chose to use a curve fit of y = m x and not y = m x + b, since

for a zero angle the acceleration should be zero. You may choose to ignore this subtlety with your students and use a simple linear fit with an intercept. The student instructions allow for either type of fit. The sample data for the cart acceleration shown here probably have a systematic error in the angle measurement.

2. The position vs. time graph is not used in analysis for this experiment; however, it can be useful for students to view it.

3. (Logger Pro and LabQuest) Seven data points are used in calculating the velocities in the data-collection program, which causes the velocity graph to be somewhat smoothed. This shows up at the end of the run, when the cart stops.

4. You will want to discuss with your students why the sin(θ) relationship is used. A vector construction showing the vector component of g along the ramp for several angles will facilitate the discussion.

ANSWERS TO PRELIMINARY QUESTIONS1. Doing the timing with one’s pulse is hard because the time between heartbeats is similar to

the time of fall. There are too few beats during the experiment.

2. It is still hard, as the experiment only lasts a few seconds. But, it is easier as the seconds hand is easier to sense than one’s pulse—and it is always consistent.

3. Answers vary.

4. It would be very hard to get useful data, but with considerable effort and repetition perhaps one could get useful data.

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Determining g on an Incline

SAMPLE RESULTS

Number of books

Height of books(cm)

Length of incline(cm)

sinθ Acceleration Average acceleration

(m/s2 )Trial 1(m/s2)

Trial 2(m/s2)

Trial 3(m/s2)

1 2.74 121.0 0.0225 0.222 0.222 0.221 0.222

2 5.18 121.0 0.0425 0.428 0.428 0.425 0.427

3 6.97 121.0 0.0572 0.572 0.571 0.571 0.571

4 9.20 121.0 0.0755 0.754 0.754 0.750 0.753

5 11.5 121.0 0.0943 0.920 0.925 0.922 0.922ANSWERS TO ANALYSIS QUESTIONS1. & 2. See data table in Sample Results.


inclineanoftopthefromreleasedcartaofvelocityandPosition


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Experiment 4

3. See graph below.

low-friction cart with a proportional fit

4. See graph below.

The extrapolated acceleration for the cart at 90° = 9.89 m/s2

5. The values are both very similar to the accepted value for the acceleration due to gravity.

6. In both cases, the values are similar to the accepted value of g. One would expect that to be the case, since if sin (θ) = 1, θ would be 90° and the track would be vertical, and the cart would fall rather than roll on the track.

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thefor(θ)sinvs.accelerationofGraph


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