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NORTHERN ILLINOIS UNIVERSITY
PHYSICS DEPARTMENT
Physics 253 – Fundamental Physics Mechanics
Lab #11 Rolling
Meet in FR 233.
Sections A, B, C, G: Tuesday Nov. 8; Sections D, E, T: Thursday Nov. 10.
Read Giancoli: Chapter 10
Lab Write-up due Tue. Nov. 15 (Sections A, B, C, G); Thur. Nov. 17 (Sections D, T)
Apparatus
In this experiment a wheel of radius R is rolled down an inclined plane with a
raised guide to keep the wheel on the track. The wheel is hollow and has partitions to
hold four steel balls. The balls can be relocated in different parts of the wheel to change
the rotational inertia I while keeping the mass m of the wheel constant.
Two photogates are positioned over the track and can measure the time, t, when
the wheel passes through the beam at each photogate. The times can be read from the
computer attached to the photogates. The photogates only record the elapsed time
between the start and stop of the beam. The experimenter must determine the width d of
the wheel at the beam to determine the velocity through each gate.
Theory
Velocity is the time rate of change of position of an object. If the width d of an
object and the time t it takes to pass a point are both known, the velocity is given by
v
d
t
(1)
Angular velocity is the time rate of change of the angular position of a rotating
object, measured in radians per second (rad/s). For an object that rolls without slipping
its linear velocity as
the angular velocity is related to

v
d

R Rt
(2)
When an object is acted upon by a force, the mass of the object provides
resistance to the force. That resistance is called inertia. When a force causes an object to
 with the lever arm (the distance away from the axis of
rotate, that force, combined
rotation that the force acts) and the direction of the force yields a torque. Resistance to
torque is not only due to mass, but also how far the mass is away from the axis of
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rotation. The resistance to rotation is called the rotational inertia. For an object of radius
R with all the mass m equally distant from an axis through the center of mass, the
rotational inertia is ICM  mR2 , where ICM is the rotational inertia about that axis through
its center of mass. For objects with the same mass distributed throughout that radius, the
rotational inertia is less. The more mass that is closer to the axis of rotation, the smaller
the rotational inertia is.

Objects in motion possess kinetic energy K. If the object is rolling it has kinetic
energy due to both the forward motion of its center of mass, KCM, and to its rotation, Krot.
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Translational kinetic energy depends on the mass and velocity, KCM  mv 2 . Rotational
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kinetic energy about the center of mass depends on the rotational inertia and angular
1
velocity, K rot  ICM  2 . The total kinetic energy is thus
2


1
1
K  mv 2  Icm 2 .
2
2
(3)
Equation (2) can be used to convert velocity in the first term to angular velocity.
 22 + ½I 2 = ½(I + mR2)2 = ½ I2.
K = ½mR
CM
CM
(4)
Notice that we have derived the parallel-axis theorem: I  ICM  md 2 where, in this case
the disk is rolling about an axis through the point where it touches the ground, thus d=R.
As a wheel rolls down a slope its potential energy due to gravity, Ug=mgh, is
converted into kinetic energy and overcomes the work done by friction, WFk. Rolling

friction is relatively small, so we shall neglect it. If the initial and final angular velocities
are  i and  f , then the change in kinetic energy is K  K f  K i . The relation between
these different forms of energy is
Ug = ½ I
(f2 – i2).
(5)
This equation [and using Eq. (4)] can be rearranged to solve for the rotational inertia
about the center of mass, ICM:
2U
ICM  2 g 2  mR2 .
(6)
 f  i
Data Collection

You must show all your calculations in the Analysis section (parts 1-6) below to your TA
before you leave the lab. This includes propagation of error calculations. Your TA will
give you the equations that relate the errors in your measured quantities to the errors in
your calculated quantities.
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(1) Open the disk and measure the mass and diameter of each ball, and measure the
mass of the empty plastic disk and the screw and fastener. Place the four steel balls
in the innermost position in the disk (closest to the center). Measure the distance of
the balls from the center of the disk. Close the disk, and measure and record the mass
m of the disk-ball system and diameter of the disk. Also record the uncertainties in
all of these numbers.
(2) Measure the height of the track at each photogate and take the difference to get the
height h and its uncertainty.
(3) Check that the computer is running the LoggerPro program.
(4) Place the wheel at the upper photogate so it just begins to interrupt the beam, as
indicated by the red light.
(5) Place a ruler at the base of the wheel along the track and push the wheel through the
gate, recording the distance di (and its uncertainty) until the wheel no longer
interrupts the beam.
(6) Repeat steps 4 and 5 for the lower photogate and record the distance df (and its
uncertainty).
(7) Start the program, release the wheel from a start position close to the first photogate,
and record the times from the computer, ti and tf
(8) Repeat the measurement from the same starting point a total of ten times. Find the
average initial and final times, t i and t f , and determine their uncertainties by
calculating the standard deviation of the mean.
(9) Repeat steps 7 and 8 using a starting point far from the first photogate.


(10)
Open the wheel and move the balls to the outer positions.
(11)
Repeat steps 7 through 9 with the second arrangement of balls.
Analysis
(1) Use the mass and radius from Step (1) to calculate mR2 and its uncertainty.
(2) Use the height from Step (2) to find the potential energy Ug=mgh and its
uncertainty.
(3) Use the average times from Step (8) and Eq. (2) to determine the initial angular
velocity  i and final angular velocity  f and their uncertainties.
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(4) Calculate the rotational inertia and its uncertainty from Equation 6.
(5) Calculate the rotational inertia and its uncertainty for the higher starting position done
in Step (9).
(6) Calculate the rotational inertia and its uncertainty for the second arrangement of balls
and for the two starting positions.
(7) What are the sources of experimental error in the measurements? Which one(s) is
most important?
(8) Do the rotational inertias measured for the same ball arrangement but different
starting positions agree to within the experimental error (that is, to within their
uncertainties)? Make this comparison for the two different ball arrangements.
(9) Was the assumption about the insignificance of friction sound? What results might
point to the occurrence of friction?
(10) From your measurements in Step (1), estimate what ICM should be for the disk-ball
system (make liberal use of the parallel-axis theorem for spheres, and use the
rotational inertia tables in the textbook). How does this compare to your measured
quantities for the two different ball orientations inside the disk? What is the
experimental difference between these two quantities? Estimate the amount of
energy converted to heat (assuming the discrepancy is all due to frictional losses).