Download Rotational Motion

Rotational Motion
Experiment Type: Cookbook
Overview
In this experiment, students will find the moment of inertia of a rotating body.
Students will observe how changing the radius at which they apply a force affects the
angular acceleration. Students will also observe how changing the magnitude of the
force affects the angular acceleration. (This lab is very similar to the regular Physics
170 experiment of the same title.)
Key Concepts
Moment of inertia, angular displacement, angular velocity, angular acceleration
Objectives
On completion of this experiment, students should be able to:
1) explain what dependence the angular acceleration has upon the radius of the
applied force
2) explain what dependence the angular acceleration has upon the magnitude of
the applied force
3) determine the moment of inertia of any rotating body
Review of concepts:
Moment of Inertia
Just what in the world is a “moment of inertia”? That’s probably the most asked
question in my lab. While it does sound a bit strange, the concept is really easy to
understand. When you study Newton’s Second Laws, F  ma, the mass is technically
called the inertial mass. You know that more inertia (more mass) means that the
object requires more force to accelerate. This is obvious because of your many
everyday experiences: You know, for example, that you are going to have to push a
 a stalled VW Beetle (a not so large
lot harder on a stalled bus (a large mass) than
mass) to get it up to the same speed. In the same way, the moment of inertia
describes how hard it is to turn an object. You will have to apply a lot more torque to
an object with large moment of inertia than you would to an object with a small
moment of inertia to get the same angular acceleration.
The moment of inertia of a body is the body's resistance to changing its rotational
motion.
Unlike mass, the moment of inertia is not a single quantity for a particular object. The
moment of inertia depends on the axis about which you want to rotate the body.
Which do you think has the greater moment of inertia, a ruler rotated about an axis
going down its length or a ruler rotated about an axis which goes perpendicular
through the face?
In our experiment, we would like to measure the moment of inertia of an object that is
composed of three cylinders of metal.
Figure 1 We wish to find the moment of inertia of this sandwich of cylinders
Of course, you can find the moment of inertia directly by measuring the mass and
radius of each cylinder, since you know that the moment of inertia is Eq. (7-1).
I   r 2 dm
(7-1)
In this equation, r is the (radial) distance (in cylindrical coordinates) from the axis of
rotation and dm is an infinitesimally small piece of mass at that distance. Solving
equation (7-1) for the sandwich of cylinders, the moment of inertia is
I
1
1
1
m1r12  m2 r22  m3 r32
2
2
2
(7-2)
In Eq. (7-2) the subscripts 1, 2, and 3 indicate the individual cylinders as labeled in
Fig. 7-1. The r1, r2, and r3 variables are the radii of the cylinders, and the m1, m2, and
m3 variables are the masses of the individual cylinders. Unfortunately, the cylinders
are stuck together, and there’s no way to measure each individual mass. However,
since we know that the cylinders are made out of the same material, we can find the
individual masses by assuming the densities of the cylinders are the same.

M total m1 m2 m3



Vtotal
V1 V2 V3
In Eq. (7-3), the variable Mtotal stands for the total mass of all three cylinders, and
Vtotal stands for the total volume of all three cylinders; V1, V2, and V3 stand for the
volume of the individual cylinders. (And, of course, I don’t need to remind you that
the volume of a cylinder is πr2L, right?). This equation can be further simplified to
find the mass of each cylinder, so that we can find the result of Eq. (7-2) directly.
(7-3)
Newton’s Second Law for Torque
You should recall that there is a “Newton’s Second Law” for torque and angular
acceleration which is very similar to the law for linear acceleration.
“Newton’s Second Law” for torque:


  I
(7-4)
Eq. (7-4) relates the net torque, , and the product of the moment of inertia, I, and the
angular acceleration, . This gives another way we can measure the moment of
inertia: we can measure the angular acceleration and we can determine the torque
applied, so we can solve for the moment of inertia.


The torque we will apply, applied, will be caused by hanging a mass, mh, on the
cylinder sandwich, as shown in Fig. 7-2.

Figure 2 A string will be wrapped around one of the cylinders. A hanging mass will be attached to the
string to apply a known force to the cylinder sandwich
Of course, friction is also present, so we need to count the frictional torque, friction.
Thus, the net torque is:
 applied   friction  I
(7-5)

If we assume that the friction is constant for all torques, Eq. (7-5) is a linear equation.
If we plot  applied vs.  , the slope should be the moment of inertia and the y-intercept
should be the frictional torque.
Torque is given by the general equation





  r F
(7-6)
so the torque on the cylinder sandwich is (when the mass is hanging from cylinder 1):
  r1 Fch
(7-7)
The force on the cylinder sandwich by the hanging mass (via the string) , Fch, is equal
in magnitude to Fhc ,the force on the hanging mass by the cylinder sandwich (via the
string) by Newton’s third law. Fhc can be computed by using Newton’s second law on
the falling mass.
Figure 3 The free body diagram of the hanging mass
Thus,
Fch  mh g  a 
(7-8)
There are then two ways to change the torque: by changing the force from the
hanging mass and by changing the radius at which we apply the torque.
In order to solve this equation, we need to know a, the acceleration of the hanging
mass. One way to determine a is to measure the amount of time it takes for the mass
to drop to the floor. Using your equations of motion, you can determine this
acceleration in terms of the distance that the mass falls, y, and the time it takes to
drop to the floor, t.
a
2y
( assuming a = constant during the fall )
t2
(7-9)
Measuring the linear acceleration of the hanging mass will also give us the angular
acceleration, since the two are connected. When the hanging mass is attached to
cylinder 1, the angular acceleration is given by Eq. (7-10).

a
r1
The coefficient of static friction is usually larger than the coefficient of kinetic
friction, so when the cylinder starts to turn, the acceleration at the beginning is
(7-10)
typically smaller, so the assumption that the acceleration is a constant may not be a
good assumption. We will use the video camcorder to record the motion and
determine whether the acceleration is constant or not. If it is not constant throughout
the falling of the hanging mass, then we will only use the portion of the fall where the
acceleration is approximately constant for determining the corresponding angular
acceleration.
Procedure
Part I: Measuring the Moment of Inertia Directly
1. Measure the radii of each of the three cylinders. Calipers are available for your
use.
2. Measure the length of each of the three cylinders. (I recommend using the back
part of the caliper for this.)
3. Determine the moment of inertia using Eq. (7-2), Eq. (7-3), and the total mass
stamped on cylinder 2.
4.
Part II: Changing the Applied Torque by Changing the Fch
1. Attach the hanging mass to one of the cylinders. Wind any excess string around
the cylinder.
2. Measure the height from the bottom of the mass to the floor. This is the distance
that the mass will drop, y,
3. Allow the mass to fall and measure the amount of time it takes for the mass to
drop to the floor.
4. In additional to step 3, also use the video camcorder to record the motion of the
falling mass (this means you have to setup a length scale, the camera, etc.)
5. Repeat step 3 & 4 at least four more times.
6. Repeat steps 1-5 at least three more times with different amounts of weights.
(You should have a whole box of weights to choose from; you decide how much
weight you want to add.)
Part III: Changing the Applied Torque by Changing the Radius
Unfortunately, there are only two radii to choose from, r1 and r3 (as defined in Fig. 7-1).
Run at least one more trial set in the same manner that you did in Part II with the other
radius (i.e. wind the string around the other cylinder) but using the same mass.
Extra credit: We assumed that the torque due to friction is constant, i.e., independent of
the applied torque, is this a good assumption? The frictional force should be proportional
to Fch. Is Fch independent of the applied torque? Analyze the problem mathematically,
find Fch and see if it is independent of the applied torque or not. Is it better to change the
applied torque by changing the hanging mass or by changing the radius as far as this
assumption is concerned?