1. Rotational Dynamics
Chapter 9

2. 9.1 The Action of Forces and Torques on Rigid Objects
What causes a change of motion?
In linear motion, we discussed the fact that the net force acting
on an object causes a change in motion. When considering rotational
motion, we need to discuss the cause of angular acceleration.
9.1 The Action of Forces and Torques on Rigid Objects
F2 F1
If both forces represent equal magnitudes, what will the difference in motion be?
α is proportional to force
α is also proportional to the perpendicular distance from the axis of rotation to the line along which the force acts
the LEVER ARM or MOMENT ARM is this perpendicular distance
Consider the following situation

3. The Lever Arm
Lever Arm – more specifically, the distance which is perpendicular both to the axis of rotation and to an imaginary line drawn along the direction of the force.
Line of action
Which force is most effective?
Why?

4. Equations for Torque
It may be conceptually easier to solve for F⊥ than for r⊥. Either way, your product will be the same because the force acts along the lever arm, so sinθ will be the same.
If more than one torque acts on a body, α is proportional to the net torque
+ counterclockwise
- Clockwise 
SI unit for torque is mN
τ = rFsinθ
τ = rF⊥
τ = r⊥F

5. 9.2 Rigid Objects in Equilibrium
A body is in equilibrium when the sum of the external forces is zero and when the sum of the external torques is zero.
In other words…∑τ = 0
Hint: When acceleration is zero the net force and in the case of rotational motion, net torque is zero.
Remember the lever arm must be determined relative to the axis of rotation.
In problem solving, choosing the direction of an unknown force backward in the free-body diagram simply means that the value determined for the force will be a negative number.

6. 9.3 Center of Gravity
The center of gravity of a rigid body is the point at which its weight can be considered to act when the torque to the weight is being calculated.
When weight is the force causing torque, the center of gravity will be the geometric center for objects with symmetrical shaped and weight distributed uniformly. Here τ = W(L/2)

7. Center of Gravity (cg) Calculations
When finding cg for more than one object, as in the diagrams, calculate the net torque created by the board and box about an axis that is picked arbitrarily to be at the right end of the board.
If the distribution of weight changes, the equilibrium may be “knocked off balance.”

8. 9.4 Newton’s 2nd Law for Rotational Motion About a Fixed Axis
Relating Newton’s 2nd Law for tangential force and acceleration to torque, we can derive a rotational motion equation for Newton’s 2nd Law.
Use rad/s2

9. Moment of Inertia of a Particle
When using F=ma, we are describing a single particle.
When using equations relating to torque, it is often convenient to describe the object rotating about an axis as a whole.
The moment of inertia can be used to describe any object rotating about a fixed axis, not just a particle.
SI unit for I is kgm2

10. More Moment of Inertia Recall that all mass has inertial properties.
The rotational analogy to mass is the moment of inertia.
The sum of the individual moments of inertia of all the particles will yield the moment of inertia of the body.
Table 9.1 on pg. 262 summarizes equations for I specific derived for specific shapes.

11. Rotational Analog of Newton’s 2nd Law Equation
α must be expressed in rad/s2

12. 9.5 Rotational Work and Energy
Definition of Rotational Work
Definition of Rotational Kinetic Energy
The rotational work Wr done by a constant torque in turning an object through an angle.
θ must be in radians
SI Unit: joule (j)
The rotational kinetic energy KER of a rigid object rotating with an angular speed ω about a fixed axis and having a moment of inertia can be derived using rotational variables.
ω must be in rad/s
SI unit: joule (j)

13. Total Mechanical Energy
Recall the law of conservation of energy can be applied to mechanical energy
In the past we used total mechanical energy as kinetic energy + potential energy.
The real story for objects experiencing rotational energy is that the total mechanical energy will be equal to kinetic energy + rotational kinetic energy + potential energy

14. Racing Cylinders!
When a hollow cylinder and a solid cylinder are rolled down an incline, the solid cylinder will reach the bottom first. This will happen because the solid cylinder will have a greater translational speed.

15. 9.6 Angular Momentum
Linear momentum can be expressed as the product of mass x velocity.
For rotational motion, the analogous concept is angular momentum (L).
Again, ω must be expressed in rad/s.
SI unit: kgm2/s

16. Conservation of Angular Momentum
The total angular momentum of a system remains constant (is conserved) if the net average external torque acting on the system is zero.
An ice skater will have a higher angular velocity when she bring her arms inward because she is decreasing her radius, thus decreasing her moment of inertia.