Download Moment of Inertia - Ryerson Department of Physics

D4-1
MOMENT OF INERTIA
OBJECTIVE
The purpose of this experiment is to verify Newton's Second Law for rotational motion; Γ = Iα,
and to measure the moment of inertia of a disc and axle.
METHOD
A heavy flywheel is set in rotation by a mass attached to a string wrapped around the axle of the
flywheel. The force exerted by the falling mass is related to the torque, Γ, and the rate of change
of angular velocity of the wheel, that is, the angular acceleration, α. The moment of inertia, I, is
the constant of proportionality between these two variables and depends on the mass and
effective radius of the rotating object. The above law can be verified by using various masses
and measuring the resulting acceleration for each as a function of the net accelerating torque.
THEORY
The magnitude of the torque is determined as the product of the
applied force and its moment arm. In this case the applied force is
the tension, T, in the string and the moment arm is the radius of the
axle, r. In order to find the tension consider the forces on the mass
m in the diagram. The net force (down) on m is F = W - T. Where
W is the weight (mg) of the mass, m. Using the letter a to represent
the downward acceleration of this mass we get the following
equation:
F = mg - T = ma
When solved for T we get:
T = mg - ma
The net torque acting on the disc and axle is thus given by:
Γ = mr(g - a)
(1)
It is clear that, in addition to other things, we must also measure the acceleration, a, in order to
determine the torque. It is actually easier to measure α, the angular acceleration, directly and
determine the linear acceleration using:
a=αr
(2)
D4-2
The angular acceleration can be calculated by determining the time, t, required for the disc to
rotate through a given angular displacement, starting from rest. For the experimental set up we
have here it will be convenient to allow the disc to accelerate through exactly eight (8)
revolutions. The average angular velocity, ωAV, for the eight turns (in radians per second) will
then be given by:
θ 8(2π ) 16 π
=
ω AV = =
t
t
t
Since the angular acceleration is constant, the final angular velocity, ωf, will be twice the
average, that is, ω f = 2 ω AV . The angular acceleration will then be given by:
α=
_ω ω f 2 ω AV 32π
=
=
= 2
t
t
t
t
(3)
The moment of inertia for a solid cylinder rotating about its axis is given by:
I = (½) MR2
(4)
where R and M are respectively the radius and mass of the cylinder. When calculating the total
moment of inertia remember the object in rotation consists of the disc and the axle each of which
is a cylinder. The moments of each part must be calculated separately and added, do not add the
masses. (Be careful to count the part of the axle inside the disc only once.)
PROCEDURE
1.
Measure all dimensions of the disc and axle (Note that the diameter of the disc is stamped
on its perimeter to 5 figures in mm). Take all other pertinent measurements so that you
can determine the moment of inertia of the disc and of the axle.
2.
Wind the string around the axle eight turns, such that, when the wheel has turned exactly
eight turns unwinding it the string will fall off. Make sure the string hangs freely - isn’t
touching wood. Suspend a reasonable amount of mass, m, on the string and measure the
time it takes after releasing the mass from rest until the string falls off (8 turns). Make
two trials and average. Record data for both trials in your data table.
3.
Repeat step 2 for five different masses.
4.
For one of the masses used, measure the time taken from the moment the string falls off
until the disc comes to rest. This may take several minutes.
5.
Use equation (3) to determine the angular acceleration corresponding to each value of m
used. Use equation (2) to determine the linear acceleration for each value of m. Use
D4-3
equation (1) to determine the torque applied to the disc and axle for each value of m.
Tabulate the results of all these calculations.
6.
Plot a graph of torque against corresponding values of angular acceleration. Determine
the slope of a best straight line representing these plotted points. Note that Lotus 1-2-3
may be useful for this part. What is the significance of this slope?
7.
Calculate the total moment of inertia of the disc and axle using equation (4) and your
measurements from procedure 1 above. In order to do this you will need to know the
density of the flywheel which is 7.87×103kg/m3 (the density of iron). Compare this value
to that obtained from the slope of the graph.
8.
Extrapolate the straight line of the graph to intersect the torque axis. The point of
intersection should correspond to a positive torque and of course a corresponding zero
angular acceleration. This non-zero torque can be interpreted as the torque required to
overcome the friction in the system. Determine this frictional torque from your graph.
9.
In step 4 of the procedure it was the frictional torque that caused the disc to slow down
and finally to stop. Use the equation Γ = Iα to determine the frictional torque. Use the
calculated value for I (step 7). The angular deceleration, α, can be determined by first
calculating the angular velocity at the instant the string fell off the axle using ω f = 2 ω AV .
Then use α = Δω/t to determine the angular deceleration. Compare the frictional torque
found here to that found in step 8.
PRELAB QUESTIONS
D4-4
1.
2.
3.
4.
[1] Define moment of inertia?
[1] What is the mathematical formula for the moment of inertia of a solid cylinder?
[1] What causes the wheel to rotate? Explain.
[1] Does the mass suspended at the end of the string fall with an acceleration equal to the
acceleration due to gravity? Explain.
5. [1] How do you relate the angular acceleration of the wheel with the linear acceleration of the
falling mass?
REPORT WRITING INSTRUCTIONS
Read the procedure on previous two pages and prepare a tables to help you record all
quantities needed for this experiment.
I. Procedure:
1. Take the pertinent measurements to determine the moment of inertia of the disc-axle directly
from its dimensions (cf. figure below)
Schematic diagram of the disc and axle
2. Determine the moment of inertia from the data recorded from the motion of the wheel.
3. Determine the torque due to friction in the bearings.
II. Conclusions:
1. How do the two values obtained for the moment of inertia compare? Give the percentage
deviation. Which value do you think is more reliable? Justify your answer.
2. Compare the two values obtained for the frictional torque. Give the percentage deviation. Which
do you consider more reliable? Justify your answer.