Physics 115: Spring 2005
Atwood’s Machine
Introduction
In this lab, you will be investigating Newton’s laws and the principle of conservation of energy.
Theory
Atwood's machine is a simple device by which Newton's laws and the conservation of
energy can be easily studied. It is a device which effectively reduces the acceleration due to
gravity. Basically, the machine consists of two weights attached to both ends of a string which
passes over a non-rotating, essentially frictionless air bearing. (See Figure 1).
The heavier of the two weights is raised to a prescribed
height h, and released from rest. By measuring the time it takes to
hit the floor, we can determine both the final velocity and
acceleration of the mass. We know that:
Figure 1
M
1
x - x0 =v0 t+ a t 2
2
h
m
describes the motion of any object undergoing constant
acceleration. If x-x0, or ∆x, is just the height h, and the initial velocity is zero, then our
acceleration can be solved for in terms of h and t.
a=
2(x - x0 )
t
2
=
2h
t
2
Our final velocity is given by
v f = v0 + at =
where we substitute 0 for v0 and 2h/t2 for a.
2h
t
1
This lab consists of measuring these quantities (the acceleration and final velocity), and
using them to test the following:
•
Using your acceleration measurement, test to see if a prediction of the acceleration
from Newton’s laws is confirmed.
•
Using your initial (0) and final velocities, and the change in height of your masses,
you can test to see if the total energy of the system (kinetic + potential for both masses) is
conserved.
Procedure
Begin with two hanging weights of equal mass. Start with a mass of about 70 to 80
grams on each side. Remember, the holders themselves have mass! Transfer a small
amount (one or two grams) of mass from one side (m) to the heavier one (M) and pull the
lighter one down so that it is just touching the floor. Adjust the height of the pulley so that
the bottom of the heavier mass is approximately 1.0 meters above the floor. Release the
weights and measure the time it takes the heavier one to hit the floor. You will need to make
sure that the drop time is long enough for you to time fairly accurately. To minimize the
timing error, stick to drop times that are larger than 2 seconds. Take care that the masses
don't swing during their upward and downward motion. (Why should this be a concern?).
Do the timing measurements three times for each trial to make sure your timing is okay.
Then take an average time for each trial. Record your timing measurements in Table 1.1A.
Before continuing, you may want to do a sample calculation (as described below) to check
that your measurements are reasonable. You can check your results with the instructor.
Transfer a little more weight (a gram or two), and repeat the procedure. Make sure that the
maximum amount of total weight transfer never exceeds about 20% of the total weight on a
side. Why is this necessary? Keeping the total mass of the system constant, again transfer a
gram or two and repeat the procedure for a total of three different mass combinations.
Record all of your data in Table 1.1A.
Now change the height of the pulley so that the heavier mass is 1.5 m above the floor.
Repeat the measurements for the same combination of masses as before. Record your data
in Table 1.1B.
Using the time measurements, calculate the final speed and acceleration of the masses.
Using the speed of the masses and the distance through which they travel, calculate the
initial and final kinetic and potential energies. In each case, calculate the energy loss
(Einitial - Efinal ) of the system. Determine a theoretical value for the final velocity assuming
energy conservation. Compare this with the measured value. Calculate a theoretical value
for acceleration using the equation you found in the pre-lab and compare this to the
experimental acceleration you calculated previously.
2
Questions
Is total energy conserved? Based on the energy difference in your results, calculate the
energy (if any) lost to friction. Remember that the lost energy is Einitial - Efinal .
In your error analysis, differentiate between systematic and random errors in the
experiment. Which do you think had the most influence? Perform a propagation of error
calculation to determine the error in your experimental values of acceleration (2h/t2) and
final velocity (2h/t).
3
Table 1.1A
Height =
Trial 1
Massheavy (M)
masslight
(m)
t1 in sec
t2 in sec
t3 in sec
tavg in sec
Final velocity vf
Ep initial
Ep final
Ek initial
Ek final
Einit - Efinal
vf (theoretical)
atheoretical
aexperimental
4
Trial 2
Trial 3
Table 1.1B
Height =
Trial 1
Massheavy (M)
masslight
(m)
t1 in sec
t2 in sec
t3 in sec
tavg in sec
Final velocity vf
Ep initial
Ep final
Ek initial
Ek final
Einit - Efinal
vf (theoretical)
atheoretical
aexperimental
5
Trial 2
Trial 3