Download AN EXPERIMENTAL STUDY OF NEWTON`S SECOND LAW

AN EXPERIMENTAL STUDY OF NEWTON'S SECOND LAW:
THE ATWOOD MACHINE
OBJECTIVES
1.
To measure the acceleration of an object that is moving with constant acceleration.
2.
To examine the experimental validity of Newton's second law.
3.
To analyze data graphically.
EQUIPMENT
Pulley, two weight holders, set of slotted weights, set of small weights including one 10g, one 5g,
five 2g, and one 1g weight, paper clips, 8 lb. test fishing line, stop watch, meter stick. Optional:
smart pulley, photogates, Science Workshop interface.
INTRODUCTION
Newton's second law states that the net external force acting on an object is directly
proportional to the mass of the object and proportional to the acceleration of the object:
∑ F = ma. We will examine the validity of this relationship by carrying out experiments to
determine the dependence of acceleration on force when the mass is kept constant. If Newton’s
second law is valid, then for an experiment in which the net force is changed while the mass is
kept constant a graph of net force vs. acceleration should be a straight line, as can be seen by
comparing the second law to the slope-intercept formula of a straight line:
∑F
y
=
=
m
m
a
x
+
+
0
b
The device used in this experiment, called an Atwood machine, consists of two masses
connected by a string that is looped over a pulley (Fig. 1). The total mass being accelerated is the
sum of m1 and m2. The frictional force fk is due to friction in the pulley. The situation is
1
equivalent to that shown in Fig. 2, from which it should be apparent that the net force acting on
the combination of two masses is
∑ F = m2g − m1g − fk.
(1)
The net force is varied without changing the total mass by transferring a small amount of mass
from m1 to m2.
In this experiment we will determine the acceleration for several different net forces. The
accelerations are found from time and distance measurements. The net force is found from
equation 1; it is necessary, therefore, to determine the magnitude of the frictional force.
PROCEDURE
A. Set up
1.
Set up the Atwood machine as shown in Fig. 1, using enough string so that when one mass
hanger is pulled to the floor the other is at least 1.5 meters above the floor.
1
The tension in the string, T, is a force exerted by each mass on the other mass. Since the tension forces are internal
to the system of two masses, they do not contribute to the net external force that accelerates the combination of the
rd
masses. (Even if one were to draw the tension forces in Fig. 2, they would cancel each other because they form a 3
law pair.)
2.
Place about 300 g on each 50 g mass holder, but arrange the masses so as to have five 2g
masses on the ascending side (m1). The system should be in equilibrium since m1 = m2.
B. Measurement of the frictional force
3.
Accurate results in this step are needed to ensure good results in the remainder of the
experiment. Add small masses (1g masspieces and/or paperclips) to m2 until the mass on
the descending side moves downward with constant velocity when given a push. Apply a
push sufficient to move the masses at a reasonable speed, i.e. not too slowly. Record m1
and the new value of m2, being sure to include the contributions from the mass holders
themselves. Use equation 1 to calculate the force of friction. (Since the system moves
with a constant velocity the net force is zero.) Do not remove the masses that you added in
this step.
C. Measurement of acceleration for various net forces
Note: We will measure the acceleration of the system using the kinematic equation
2
y − yo = vot + at /2. If the system starts from rest, then a measurement of the time taken for
the system to travel a known distance will permit the acceleration to be calculated. If this
method is used, it is a good idea to make several measurements of the time interval.
Alternatively, the acceleration can be determined by using a computer-interfaced “Smart
Pulley” to record a graph of v vs. t. The slope of v vs. t is the acceleration. A procedure
for the first method is given below.
4.
5.
6.
Transfer 2 g from m1 to m2 to create a nonzero net force on the system. (What is the value
of the net force?)
Starting with m1 on the floor, measure the amount of time required for m2 to fall to the
floor, starting the stopwatch the instant you release m1. Make and record four independent
timings. Record m1 and m2. Measure the distance traveled by the ascending mass during a
timing.
Increase the net force by transferring an additional 2 g from m1 to m2. Repeat step 5.
Repeat this procedure until all five of the 2 g masspieces have been transferred from m1 to
m2 .
DATA ANALYSIS
A. Determination of acceleration
The acceleration of the system was not measured directly, but we can calculate it using our
data and the kinematic equations for motion with uniform (constant) acceleration. The
pertinent equation is
2
y − yo = vot + at /2,
1.
2.
(2)
where y is the distance traveled by the system in time t. In this experiment the masses were
started from rest, so vo = 0
Calculate the average time of travel for each of the net accelerating forces.
Use equation 2 to calculate the acceleration produced by each net force.
B. Examination of Newton's second law
3.
Use equation 1 to calculate the net force corresponding to each acceleration.
4.
According to Newton's second law, ∑F = ma, a plot of ∑F vs. a should be a straight line.
Graph ∑F vs. a, and perform a linear regression to determine the equation of the best
2
5.
straight line through the points. Report the slope and y-intercept with the proper units and
the proper number of significant figures. Do your experimental points fall on a straight
line?
Are the values of the slope and y-intercept equal, within experimental uncertainty, to those
predicted by the second law? Compare your experimental values to the predicted values.
(Of course, you should include a percent difference in your comparison.) If the slope of
your graph did not equal m1 + m2, try to think of sources of error in the experiment. For
example, did we take account of all the mass that was being accelerated?
fk
T
T
m1
a
m2
m1g
m2g
Figure 1.
The Atwood Machine. Forces acting of masses 1 and 2 are shown.
a
fk
m1g
m2g
m1
m2
Figure 2.
A linear analog of the Atwood Machine. Forces that act on the combination of two masses are
shown.
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