1. Today: (Ch. 8)
 Rotational Motion

2. Center of Gravity
For the purposes of calculating the torque due to the gravitational force, you can assume all the force acts at a single location
The location is called the center of gravity of the object
The center of gravity and the center of mass of an object are usually the same point

3. Rotational Equilibrium
Equilibrium may include rotational equilibrium
An object can be in equilibrium with regard to both its translation and its rotational motion
Its linear acceleration must be zero and its angular acceleration must be zero
The total force being zero is not sufficient to ensure both accelerations are zero

4. Equilibrium Example
The applied forces are equal in magnitude, but opposite in direction
Therefore, ΣF = 0
However, the object is not in equilibrium
The forces produce a net torque on the object
There will be an angular acceleration in the clockwise direction
For an object to be in complete equilibrium, the angular acceleration is required to be zero
Στ = 0
This is a necessary condition for rotational equilibrium
All the torques will be considered to refer to a single axis of rotation
The same ideas can also be applied to multiple axes

5. Rotational Equilibrium, Lever
Use rotational equilibrium to find the force needed to just lift the rock
We can assume that the acceleration is zero
Also ignore the mass of the lever
The force exerted by the person can be less than the weight of the rock
The lever will amplify the force exerted by the person
If Lperson > Lrock

6. Tipping a Crate
We can calculate the force that will just cause the crate to tip
When on the verge of tipping, static equilibrium applies
If the person can exert about half the weight of the crate, it will tip

7. Moment of Inertia
The moment of inertia composed of many pieces of mass is
The moment of inertia of an object depends on its mass and on how this mass is distributed with respect to the rotation axis
The definition can be applied to find the moment of inertia of various objects for any rotational axis
Units of moment of inertia are kg · m2

8. Various Moments of Inertia

9. Rotational Dynamics Newton’s Second Law for a rotating system states
Once the total torque and moment of inertia are found, the angular acceleration can be calculated
Then rotational motion equations can be applied
For constant angular acceleration:

10. Kinematic Relationships

11. Real Pulley with Mass, Example
Up to now, we have assumed a massless pulley
Using rotational dynamics, we can deal with real pulleys
The torque on the pulley is due to the tension in the rope
Apply Newton’s Second Laws for translational motion and for rotational motion
The crate undergoes translational motion
The pulley undergoes rotational motion
For the pulley:
The tension in the rope supplies the torque
The pulley rotates around its center, so that is a logical axis of rotation

12. Motion of a Crate, Example, cont
Pulley equation
Στ = - T R = Ipulley α
The pulley is a disc, so I = ½ mpulley R²pulley
For the crate
Take the +y direction as +
Equation: ΣF = T – mcrate g = mcrate a
Relating the accelerations
a = α Rpulley
Combine the equations and solve

13. Example
If the mass of a wheel is increased by a factor of 17 and the radius is increased by a factor of 9, by what factor is the moment of inertia increased?
A factor of

14. Example
If the mass and height of the object is increased by twice, what would be the increase/decrease of final potential energy?

15. Example
If final velocity is increased by 3 times, what would be increase/decrease in final kinetic energy?

16. Tomorrow: (Ch. 9)
 Energy and Momentum of Rotational Motion