1. Center of Mass & Rotational Inertia

2. AP Physics 1 Bell Ringer – 3/13
Why do SUVs tip over more easily than cars?

3. Center of Gravity (Center of Mass)
The center of gravity of an object is the point at which all the weight of an object is considered to be acting (for purposes of treating forces and torques that affect the object).
The single support force has line of action that passes through the c. g. in any orientation.

4. Examples of Center of Gravity
Note: C. of G. is not always on the object.
Bozeman Video on Center of Mass

5. Center of Gravity Activities
Stand up from chair.
Pick up a chair.
Pick up a coin from the floor.
Lift your leg.
Try all of the above with a backpack.

6. Bell Ringer
Make a Prediction: Which object in the following picture would you predict would reach the bottom of the incline first: the solid cylinder or the thin ring? Both objects have the same mass and equal diameters. (See solution in a few slides.)

7. Rotational Inertia What is inertia?
An object’s ability to resist changes in motion.
Rotational inertia refers to an object’s ability to resist changes in rotational motion.
Two things effect an object’s ability to resist rotational motion.
The object's mass
How far away that mass is from the center of rotation.
A property called moment of inertia measures how mass is distributed around a center of rotation.

8. Does an object’s shape matter?
Here are the three most popular rolling objects.
Notice that the rotational inertia increases from left to right…
What could be the reason for this?
Answer: Rotational inertia increases as the mass distribution gets farther from the axis of rotation.
thin-walled cylinder/hoop/ring (about central axis)
I = MR2
solid cylinder/disk  (about central axis)
I = 1/2 MR2
solid sphere I = 2/5 MR2
as the mass distribution gets farther from the axis of rotation that passes through their center of mass.

9. But, how do we figure out the rotational inertia of something?
For a point particle that is moving in a circle around an axis, its rotational inertia is given by the equation I = mr2.
For an object with some other shape (like a ball, disk, rod, etc) a formula will be given if you need it. (See next slide for rotational inertias of some common shapes.)
3. For a system consisting of several objects, you can add together the rotational inertias of each object to find the total rotational inertia of a system. Note: You DO NOT need to memorize the rotational inertias of any of the shapes.

10. Common Rotational Inertias
I = mR2
I = ½mR2
Hoop
Disk or cylinder
Solid sphere

11. The solid cylinder gets to the bottom first!
Bell Ringer
Which object in the following picture would you predict would reach the bottom of the incline first: the solid cylinder or the thin ring? Both objects have the same mass and equal diameters.
The solid cylinder gets to the bottom first!
WHY?
The cylinder has a smaller moment of inertia and will therefore accelerate faster down the ramp.
To see this result, view this video.

12. Bell Ringer
List the types of kinetic energy and potential energy that we have discussed.

13. Angular Momentum Defined
Consider a particle m moving with velocity v in a circle of radius r.
Let’s derive Angular Momentum, L:
Start with:
Since I = mr2, we have:
L = (mv)r
L = Iw
Substituting v= wr, gives:
L = m(wr) r = mr2w
Angular Momentum
For a rotating body:
L = (mr2)w

14. Example 8: Find the angular momentum of a thin 4-kg rod of length 2 m if it rotates about its midpoint at a speed of 300 rpm.
m = 4 kg
L = 2 m
For rod: I = mL2 = (4 kg)(2 m)2
1 12
I = 1.33 kg m2
L = Iw = (1.33 kg m2)(31.4 rad/s)2
L = 1315 kg m2/s

15. Impulse and Momentum
Recall for linear motion the linear impulse is equal to the change in linear momentum:
Using angular analogies, we find angular impulse to be equal to the change in angular momentum:

16. Impulse = change in angular momentum
Example 9: A sharp force of 200 N is applied to the edge of a wheel free to rotate. The force acts for s. What is the final angular velocity? R
2 kg w F
wo = 0 rad/s
R = 0.40 m
F = 200 N
D t = s
I = mR2 = (2 kg)(0.4 m)2
I = 0.32 kg m2
Applied torque t = FR
Impulse = change in angular momentum
t Dt = Iwf - Iwo
FR Dt = Iwf
wf = 0.5 rad/s

17. Conservation of Momentum
In the absence of external torque the rotational momentum of a system is conserved (constant).
Ifwf = Iowo
Ifwf - Iowo = t Dt
Io = 2 kg m2; wo = 600 rpm
If = 6 kg m2; wo = ?
wf = 200 rpm

18. Summary – Rotational Analogies
Quantity
Linear
Rotational
Displacement
x, y, d or s (m)
 (radians)
Inertia
Mass (kg)
I (kgm2)
Thing the causes an acceleration
Force (N)
Torque (N·m)
Velocity
v (m/s)
 (rad/s)
Acceleration
a (m/s2)
 (rad/s2)
Momentum
mv (kg m/s)
I (kgm2rad/s)

19. = Summary of Formulas: I = SmR2 mgho mghf Height? ½Iwo2 ½Iwf2
½mvo2 =
mghf
½Iwf2
½mvf2
Height?
Rotation?
velocity?

20. Bell Ringer
When an ice skater spins they start out and end with their arms out. Then they bring their arms in.. What does this do to their motion? Why?
It makes them spin faster!
This is why:
When they bring their arms in, they are changing the shape of their body – this new “shape” has a smaller rotational inertia, so the angular velocity must be proportionally larger.

21. WIND TURBINES such as these can generate significant energy in a way that is environ-mentally friendly and renewable. The concepts of rotational acceleration, angular velocity, angular displacement, rotational inertia, and other topics discussed in this chapter are useful in describing the operation of wind turbines.