1. Dynamics of a Rigid Body

2. General Definition of Torque
Let F be a force acting on an object, and let r be a position vector from a rotational center to the point of application of the force. The magnitude of the torque is given by
 = 0° or  = 180 °:
torque are equal to zero
 = 90° or  = 270 °: magnitude of torque attain to the maximum

3. Torque Units and Direction
The SI units of torque are N.m
Torque is a vector quantity
Torque magnitude is given by
Torque will have direction
If the turning tendency of the force is counterclockwise, the torque will be positive
If the turning tendency is clockwise, the torque will be negative
The work done by the torque is given by

4. Net Torque
The force will tend to cause a counterclockwise rotation about O
The force will tend to cause a clockwise rotation about O
St = t1 + t2 = F1d1 – F2d2
If St  0, starts rotating
If St = 0, rotation rate does not change
Rate of rotation of an object does not change, unless the object is acted on by a net torque

5. Power delivered by torque
To find the instantaneous power delivered by the torque, divide both sides by dt or

6. Newton’s Second Law for a Rotating Object
When a rigid object is subject to a net torque (≠0), it undergoes an angular acceleration
The angular acceleration is directly proportional to the net torque
The angular acceleration is inversely proportional to the moment of inertia of the object
The relationship is analogous to

8. Extended Work-Energy Theorem
The work-energy theorem tells us
When Wnc = 0,
The total mechanical energy is conserved and remains the same at all times
Remember, this is for conservative forces, no dissipative forces such as friction can be present

9. Total Energy of a System
A ball is rolling down a ramp
Described by three types of energy
Gravitational potential energy
Translational kinetic energy
Rotational kinetic energy
Total energy of a system

10. Conservation of Mechanical Energy
Remember, this is for conservative forces, no dissipative forces such as friction can be present

11. Work-Energy in a Rotating System
The work done on the body by the external torque equals the change in the rotational kinetic energy
The work equals the negative of the change in potential energy
Conservation of Energy in Rotational Motion

12. Problem Solving Hints Choose two points of interest
One where all the necessary information is given
The other where information is desired
Identify the conservative and non-conservative forces
Write the general equation for the Work-Energy theorem if there are non-conservative forces
Use Conservation of Energy if there are no non-conservative forces
Use v = w to combine terms
Solve for the unknown

13. A meterstick is initially standing vertically on the
Example
A meterstick is initially standing vertically on the
floor. If the falls over, with what angular velocity
will it hi the floor?
Moment of inertia is Ml2/2
l = 1.0 m
U = mgy y0 = com = .5 m
ω0 = 0
y0 = com = 0 m
Therefore,
What rate is gravity delivering energy? The mass of the meterstick is 0.15kg
R= l/2

14. General Problem Solving Hints
The same basic techniques that were used in linear motion can be applied to rotational motion.
F becomes 
m becomes I
a becomes 
v becomes ω
x becomes θ

15. Angular Momentum
Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum
Linear momentum is defined as
Angular momentum is defined as

16. Angular Momentum and Torque
Net torque acting on an object is equal to the time rate of change of the object’s angular momentum
Angular momentum is defined as
Analog in impulse

17. Angular Momentum Conservation
If the net torque is zero, the angular momentum remains constant
Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero.
That is, when
then

18. Blocks and Pulley
Two blocks with masses m1 = 5 kg and m2 = 7 kg are attached by a string over a pulley with mass M = 2kg. The pulley, which runs on a frictionless axle, is a hollow cylinder with radius 0.05 m over which the string moves without slipping.

20. Blocks and Pulley
Two blocks with masses m1 = 5 kg and m2 = 7 kg are attached by a string over a pulley with mass M = 2kg. The pulley, which runs on a frictionless axle, is a hollow cylinder with radius 0.05 m over which the string moves without slipping.

21. An automobile with rear-wheel drive is accelerating at 4
An automobile with rear-wheel drive is accelerating at 4.0m/s2 along a straight road. Consider one of the wheels of this automobile. The axle pushes forward, providing an acceleration of 4.0 m/s2. Simultaneously, the friction force of the road pushes the bottom of the wheel backward, providing a torque that gives the wheel an angular acceleration. The wheel has a radius of 0.38m and a mass of 25kg. Find the backward force that the friction force exerts on the wheel, and find the forward force that the axle exerts on the wheel.