N otes on Chapter 8
pp 73 - 91 of your lab book

Since we rarely deal with "point" particles in real life, we must strive to understand rotational motion. Unlike point particles, real objects (two or three dimensional objects) will spin or turn when an off-center force influences them. Such an "off-center force" is called a torque. It is a torque which we apply to a wrench to turn a nut, and a torque we apply to pedal our bicycles.

The next thing you must understand in order to learn about rotational motion is something called the moment of inertia. 

• The moment of inertia of a body is the body's resistance to changing its rotational motion.

This is similar to ordinary inertia (mass) which is the amount of resistance to changing linear motion (accelerating). The bigger moment of inertia, the harder it is to increase or decrease that body's rotational speed, just like a larger mass means that it is harder to change linear acceleration. But unlike mass, the moment of inertia is not a set quantity for a particular object. The moment of inertia depends on the axis about which you want to rotate the body.
 
Class participation: Which to 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?    

Once you have mastered these concepts, learning the rest of rotational kinematics is easy. Every equation in rotational kinematics has a linear analog which you already know, and it is easy to go from one to the other. (You may want to save this table for your lecture course! Note: 170 students please replace delta with infinitesimal change, d, such that v = dx/dt)

Rotational equation or concept	Linear Analog	Conversion from linear to rotational
moment of inertia (I)	mass (m)
torque ( ) = I	force (F) F = ma	= r x F (where r is the distance from the center of mass to where the force is being applied)
angular displacement ( )	linear displacement (s)	(where s = the arc distance)
angular velocity ( )	linear velocity (v)
angular acceleration ( )	linear acceleration (a)
angular momentum (L) L = I	linear momentum (p) p = mv
rotational kinetic energy (Krot) K = 1/2 I 2	linear kinetic energy (K) K = 1/2 mv2

Now to the situation in our lab. We have three metal cylinders of different radii attached to one another. This is the object which we will be rotating. We will apply a torque to this object to get it spinning.

The torque we will apply ( _applied) will be caused by hanging a mass (m) on the cylinder. Another torque is caused by friction ( _friction).

net torque = _applied  - _friction = I

Notice that this equation is a linear eqaution.  If you plot  _applied vs. then the slope of this graph is the moment of intertia (I) and the y intercept is the frictional torque ( _friction).

What we need to find is the applied torque and the angular acceleration. The applied torque = r x F_applied, as shown in the table above. That means that a larger torque will be applied when the radius of the cylinder is greater. In our lab we will apply four different torques by changing the radius and by changing the applied force. We can find the applied force easily. The applied force is the tension (T) in the string. The tension is given by (see free body diagram on page 75):

mg  - T = ma

In order to solve this equation, we need to know a, the acceleration of the mass. To do this, we can measure the amount of time it takes for the mass to drop to the floor. Using your equation of motion, you know that:

a = 2y/t ²

where y is the height that the mass falls, and t is the time that it takes for the mass to frop to the floor.

Measuring a will also give us the angular acceleration, since
 

 =a /r

where r is the radius of the cylinder to which the mass is attached.

P rocedure:

1. Measure the radii of the small axle, the larger axle, and the wheel as labeled in the diagram to the right. (You can use the caliper to measure the diameter and divide by two to get the radius.)
2. Measure the thickness of the cylinders. Your data table calls this the "length of the axle". Log all of this on the data tables on pages 83 and 85.
3. The mass of the entire apparatus (consisting of the wheel, the small axle, and the larger axle) is stamped on the wheel. Be sure to note this mass on pg. 85.
   
4. Tie a 50 g mass to the small axle. Wind any excess string around the small axle.
5. Measure the height from the bottom of the mass to the floor (as shown in the diagram).
6. Measure the amount of time it takes for the mass to hit the floor.
7. Repeat step 6 four more times so you can find the SD and SDM for the time measurements.
   
8. Repeat steps 4-7 with the larger cylinder.
9. Repeat steps 4-8 for the 100 g mass.
10. Plot the applied torque versus the angular acceleration

(we will not be using the computerized version of this. )

Data: Data tables are located in your book on pg. 81, 83, and 85. Please tear them out and paste them in your lab book .