3. Atwood's Machine Experiment

In performing this experiment, you will learn about three important aspects of experimental physics. The

apparatus is technically more sophisticated than the ones used in the previous experiments. You will

measure a time interval with an electronic timer that automatic starts and stops. Atwood's machine is a

device which allows a kinematic measurement of the acceleration of gravity by slowing the motion of a

pair of weights. There are many possible causes of systematic error which you will have to be on the

lookout for. Finally, you will learn to linearize the mathematical expression relating the dynamic variables

describing the motion so that the experimental result can be obtained by fitting a straight line to the data.

This is an important method of data analysis, which you will use repeatedly during this course.

Purpose

To determine the local acceleration due to gravity by means of Atwood's Machine and measure the

frictional torque of the apparatus.

Apparatus

The Atwood Machine consists of two unequal weights joined by a light inextensible string which passes

over a light pulley. A solenoid magnet and micro-switch have been added to the machine to provide a

precise method of timing the movement of the weights.

Figure 1: Atwood's Machine Apparatus

Theory

Assume 𝑚2 > 𝑚1. Applying the second law of motion on weights

𝑚2𝑔 − 𝑇2 = 𝑚2𝑎 (3.1)

and 𝑇1 − 𝑚1𝑔 = 𝑚1𝑎 (3.2)

Figure 2: Atwood's machine forces diagram

In the equations of motion for rotating systems, friction always appears in the form of a torque (or moment)

which tends to slow down the rotation; there is a counter-clockwise torque 𝜏 = 𝑇2′𝑟- 𝑇1′𝑟, where 𝑇1′ and

𝑇2′ are the reactions to 𝑇1 and 𝑇2, having the same magnitudes (𝑇𝑖′ = 𝑇𝑖) and 𝑟 is the radius of the pulley.

This torque causes the pulley to rotate with angular acceleration, 𝛼 and for the rotation of the pulley

(torque equation)

𝐼𝛼 = 𝑇2𝑟 − 𝑇1𝑟 − Γ (3.3)

where 𝐼 is rotational inertia of the pulley, Γ is the torque due to the friction in the axel, and 𝑟 is again the

radius of the pulley. We also assume that the angle between the string and the radius of the pulley is always

𝜃 = 90𝑜 so that sin(𝜃) = 1.

Now, one needs a fourth equation (unknowns are 𝑎, 𝛼, 𝑇1, 𝑇2) and it comes from our assumption that there

is no slipping of the string over the pulley. So,

𝛼 = 𝑎

𝑟 (3.4)

Substitution of 𝛼 into the torque equation gives

𝐼𝑎

𝑟2+

Γ

𝑟= 𝑇2 − 𝑇1 (3.5)

Adding the first two equations gives

𝑚2𝑔 − 𝑚1𝑔 − (𝑇2 − 𝑇1) = 𝑎(𝑚2 + 𝑚1) (3.6)

Combining Eq. 3.5 and 3.6 results in

𝑎 =(𝑚2 − 𝑚1)𝑔 − Γ

𝑟⁄

𝑚1 + 𝑚2 + 𝐼𝑟2⁄

(3.7)

Or, in order to use only total mass 𝑀 = 𝑚1 + 𝑚2, and the mass difference Δ𝑚 = 𝑚2 − 𝑚1 the

acceleration looks like this

𝑎 =Δ𝑚 𝑔 − Γ

𝑟⁄

𝑀 + 𝐼𝑟2⁄

(3.8)

Since the masses move along straight, vertical lines, at a constant accelerations, one can determine

acceleration by measuring the time, 𝑡, it takes for 𝑚1 to rise a distance ℎ from the rest using relation

ℎ =𝑎𝑡2

2 (3.9)

Eliminating 𝑎 from last two equations, a cumbersome relation appears

1

𝑡2=

𝑔

2ℎ(𝑀 + 𝐼/𝑟2)Δ𝑚 −

Γ

2ℎ𝑟(𝑀 + 𝐼/𝑟2) (3.10)

Now, this is a linear equation 1

𝑡2 vs Δ𝑚. One needs then to measure, and plot,

1

𝑡2 vs Δ𝑚 with a sufficient

number of points and then from the slope and the 𝑦-intercept, 𝑔 and Γ could be calculated.

Procedure

Mass measurement

1. The weights provided are two larger weights (𝑚 ≈ 250 g) and 10 washers (𝑚𝑤 ≈ 1 g).

2. Weigh the weight with the iron insert, including the screw. This weight has a rusted circle under

it, it’s the only one that will be attracted to the magnet. This will be 𝑚1. Let the string hang on the

side and make sure it doesn’t pull on the scale; the mass of the string will be neglected.

3. Weigh the other weight with its screw, this one is 𝑚2.

All measurements will have to be performed three times. The variation of the measured quantity will

indicate the statistical error which will have to be compared with the reading error of your instrument.

4. Measure the total mass of ten washers, a division by 10 gives the mass of one washer 𝑚𝑤. The

error (statistical or reading) will also have to be divided by 10.

5. Enter your data in a table similar to Table 1, in your notebook.

Masses Lengths

Trial # Weight

with the

iron core+

screw

Other

weight +

screw

Mass of

one washer

(mass of

all/10)

distance

travelled by

𝑚1

Radius of the

pulley

(diameter/2)

𝑚1 (𝑔) 𝑚2 (𝑔) 𝑚𝑤 (𝑔) ℎ (𝑐𝑚) 𝑟 (𝑐𝑚)

1

2

3

Average

Reading error

Standard Deviation (𝜎)

Std Dev mean (𝜎𝑚𝑒𝑎𝑛)

Final Measurement

(average ± error) units

Table 1: Example of a table for masses and lengths

Set up of apparatus

6. Thread the string, with the masses at both ends, over the pulley. The string must pass through the

guide holes on both sides of the pulley. The mass with iron insert must be on the solenoid side,

and initially, the ten washers must be on the other side. The string may slip, and the weights may

hit you, so be careful.

7. Measure the travelling distance of 𝑚1 (ℎ) from the top of the solenoid to the bottom of the mass

when it is in contact with the micro-switch at the top.

8. In order to determine the radius of the pulley (𝑟), measure from the separation of the two sides of

the string. Try to be as accurate as possible; it might help to put the same mass on both sides. Then

divide by 2 since you measured the equivalent of the diameter and , 𝑟 =𝑑

2. You’ll also need to

divide the reading error by 2.

9. Set the POWER switch, on the front panel of the clock, to ON. The mass should be kept in place

by the electromagnet.

10. At first you will have 10 washer on 𝑚2 and 0 on 𝑚1. Then, 9 washers on 𝑚2 and 1 on 𝑚1. All

the way down to 6 washers on 𝑚2 and 4 on 𝑚1. For each ∆𝑚 make enough trials to have five

consistent times, {𝑡1, 𝑡2, 𝑡3, 𝑡4, 𝑡5}. Proceed as follows:

a. Carefully raise 𝑚2 until 𝑚1 is in contact with the solenoid, and then, RESET the clock.

This will cause the solenoid to be activated, thus, holding 𝑚1 in place. This procedure gives

minimum oscillations of 𝑚1 as it rises.

b. Having checked that everything is in order, press the START button (or the trigger switch).

This action releases 𝑚1 and starts the clock simultaneously. The clock automatically stops

when 𝑚1 hits the micro-switch at the top. Make sure that the string doesn’t swing like a

pendulum, otherwise it will introduce a centripetal force and the string might rub against

the apparatus.

11. Enter your data in the Logger pro file found on Culearn.

12. Once you have five successful trials, transfer one washer from the 𝑚2 side to the 𝑚1 (solenoid)

side. Note that when you transfer one washer from one side to the other, the difference of mass

(∆𝑚) increase by 2𝑚𝑤. Recheck that the string is properly threaded in the guide holes and that

alignment is good before proceeding with the next point. Measure for all five points (10-0, 9-1, 8-

2, 7-3, 6-4).

Data Analysis

13. The Logger Pro file will complete several columns automatically, others will have to be added by

hand. You have to provide us with a sample calculation in your lab report, even for the

columns that are automatically calculated!

14. Calculate the difference of mass between the two sides (∆𝑚) for each configuration, for

example:

1st ∆𝑚10−0 = (𝑚2 + 10𝑚𝑤) − 𝑚1

2nd ∆𝑚9−1 = (𝑚2 + 9𝑚𝑤) − (𝑚1 + 𝑚𝑤)

3rd ∆𝑚8−2 = (𝑚2 + 8𝑚𝑤) − (𝑚1 + 2𝑚𝑤)

4th …

5th …

15. The error on ∆𝑚 have to be calculated by combining three measurements together (using both the

addition and subtraction error propagation rules). This equation is only good for the ∆𝑚10−0, but

for simplicity, we will use it for the other four 𝜎∆𝑚:

𝜎∆𝑚 = √𝜎𝑚12 + 𝜎𝑚2

2 + 100𝜎𝑚𝑤2 (3.11)

16. The clock can be read down to 0.001 s, but you may find that the times for a given ∆𝑚 are not

reproduced with this accuracy.

a. Fill the Reading Error (RE) column.

b. It’s more precise to calculate the average time (𝑡𝑎𝑣) for your 𝑁 measurements 𝑡𝑖,

𝑡𝑎𝑣 = ∑ 𝑡𝑖

𝑁 (3.12)

WARNING: When adding or removing washers, rest the OTHER weight on the bench top.

Failure to follow this procedure can lead to damage or injury.

c. Since you only have a few measurements you will need to use the inefficient statistics

equation for the standard deviation 𝜎.

𝜎 =𝑡𝑚𝑎𝑥 − 𝑡𝑚𝑖𝑛

√𝑁 (3.13)

𝜎𝑚𝑒𝑎𝑛 =𝜎

√𝑁 (3.14)

d. Logger Pro will automatically take 𝜎𝑚𝑒𝑎𝑛 as the dominant source of error, which is right

for most of your measurements. Nevertheless, it is your responsibility to compare the

reading error of the clock with the standard deviation just like before:

𝜎 ≤ 2 × the Reading error, use the Reading error (3.15)

𝜎 > 2 × the Reading error, use 𝜎𝑚𝑒𝑎𝑛 (remember 𝜎𝑚𝑒𝑎𝑛 =𝜎

√𝑁) (3.16)

and pick either the reading error or the statistical error as your dominant source of error. It

is always good practice to specify which error was the dominant one for each measurement.

In the calculation section you need to show how you compared the two sources of error, at

least once.

17. From 𝑡𝑎𝑣, you can calculate 1

(𝑡𝑎𝑣)2 since you’ll need this to draw your graph. You’ll also need to

find the error on this quantity which gives the vertical error bars for experimental points. If you

use error propagation techniques, you should find:

𝜎 1

𝑡𝑎𝑣2

=2

𝑡𝑎𝑣3 𝜎𝑡𝑎𝑣

(3.17)

18. The corresponding x-coordinates are ∆𝑚, that you already calculated. All horizontal error bars will

have the same size.

It is very useful to calculate the plotted quantities and to plot them as you make the measurements. This

is the only way you can become aware of inconsistencies in your data. It is not good to discover

measurement errors after the apparatus has been dismantled and you have left the lab.

Plotting the graph

19. If you take 1

𝑡𝑎𝑣2 vs Δ𝑚, you should obtain a linear relationship. Using logger pro, fill all columns

and the graph should appear with error bars on each points. Make sure that the title is correct (to

add a title, double click on the graph), check your units and the scale of the graph.

20. Find the slope (𝑚) and the error on the slope (𝜎𝑚) by fitting a straight line (Analyze – Linear Fit).

If the error on the slope does not appear, double click on the Linear fit panel and check “Show

Uncertainty”. Make sure that the font in the Linear Fit panel is big enough.

21. To calculate 𝑔 and its error (𝜎𝑔), it’s easier if we define A and its error (𝜎𝐴), as

𝐴 = 𝑀 + 𝐼𝑟2⁄ (3.18)

𝜎𝐴 = √𝜎𝑀2 + 𝜎𝐼/𝑟2

2 (3.19)

22. Remember that 𝑀 is the total mass on the machine, you need to calculate it. Calculate also its error.

𝑀 = 𝑚1 + 𝑚2 + 10𝑚𝑤 (3.20)

𝜎𝑀 = √𝜎𝑚12 + 𝜎𝑚2

2 + 100𝜎𝑚𝑤2 (3.21)

23. The quantity 𝐼/𝑟2 is related to the inertia of the pulley, for your apparatus, the given value is:

𝐼/𝑟2 = (80 ± 1) 𝑔 (3.22)

24. Then, by using Eq. 3.10, you find the slope equal to:

𝑚 =𝑔

2ℎ(𝑀 + 𝐼/𝑟2 )=

𝑔

2ℎ𝐴 (3.23)

and therefore, you can solve for 𝑔:

𝑔 = 2𝑚ℎ𝐴 (3.24)

Then the error on 𝑔 is:

𝜎𝑔 = 𝑔√𝜎𝑚

2

𝑚2+

𝜎ℎ2

ℎ2+

𝜎𝐴2

𝐴2 (3.25)

25. To find the frictional torque (Γ), you need the y-intercept (𝑏). By setting ∆𝑚 = 0 in Eq. 3.10, the

y-intercept is:

𝑏 = −Γ

2ℎ𝑟𝐴 (3.26)

It is simply a matter of rearranging the last equation to calculate Γ and its error

Γ = −2𝑏ℎ𝑟𝐴 (3.27)

𝜎Γ = Γ√𝜎𝑏

2

𝑏2+

𝜎ℎ2

ℎ2+

𝜎𝑟2

𝑟2+

𝜎𝐴2

𝐴2 (3.28)

Notes

Present your results as 𝑔 = ( … ± . . . )𝑚

𝑠2 and Γ = ( … ±. . . )𝑁𝑚 . For 𝑔, to get SI units it is enough to

express ℎ in meters, however for Γ you will need to also express masses in kg .

Don’t forget to write a meaningful discussion. If you run out of ideas, expand these topics:

How does your value of 𝑔 compare with the accepted value (9.81𝑚

𝑠2)?

What could you do to increase the precision of your measurements? (which instrument to replace

or what procedure to follow)

Is the sign of Γ correct? Explain.

How does this experiment compares with one where you would calculate 𝑔 from the time it takes

for an object for falling on the floor (think about your reaction time).

Is your measurement precise enough to find a variation of g of ~0.00005 m/s2 (like over a oil

deposit)?

Is friction part of the error in the measurement of 𝑔? Why?

What about air resistance? Is it a significant source of error? Why?

What is the effect the weight of the string? And of its elasticity?