When the axis of rotation is fixed, all particles move in a circle. Because the object is rigid, they move through the same angular displacement in the same time period. Radians only! Note that as long as there is rotation there will be a radial or centripetal acceleration given by

1-D linear motion Rotation: fixed axis ∆x∆x∆ v a m F F=ma p=mv P=Fv Linear 1-D vs Fixed Axis Rotation

Constant Angular Acceleration Kinematics The equations for 1-D motion with constant acceleration are a result the definitions of the quantities; because it immediately follows that if acceleration is constant. Since the angular variables θ, ω, α are related to each other in exactly the same way as x, v, and a are, it follows that they will obey analogous kinematic equations: Exercise: A disc initially rotating at 40 rad/s is slowed to 10 rad/s in 5 s. Find the angular acceleration and the angle through which it turns during this time.

Moment of inertia about the axis of rotation

Moment of Inertia Examples LCR

Continuous Matter Distribution Example: Uniform rod, length L, mass M. Find I Left 1) Find contribution of arbitrary little piece dm. 2) Sum up (integrate over correct limits) all the little pieces.

Uniform disc, radius R, mass M. Find I C 1) Find contribution of arbitrary little piece dm. All the mass in ring is same r away. But what is dm? The area in the ring is an infinitesimal: Let σ designate mass per area: Then 2) Sum up (integrate over correct limits) all the little pieces.

Parallel Axis Theorem Total mass

Example: Find I CM for a uniform stick of length L, mass M. L Exercise: Find I about a point P at the rim of a uniform disc of mass M, radius R. P

Torque From Newton’s 2 nd law we know that forces cause accelerations. We might ask what particular quantity, obviously related to force, will cause angular accelerations. Consider a 10 N force applied to a rod pivoted about the left end. We can apply the force in a variety of ways, not all causing the same angular acceleration: From the examples it should be clear that it is a combination of the force applied, direction of application, and distance from axis that need to be accounted for in determining if a force can cause angular acceleration.

The previous discussion is consistent with focusing on the following quantity: R Line of action of F R eff or lever arm The greater the torque associated with a force, the more efficient it is in creating angular acceleration.

Torque on piece Refers to a particular axis

H Example: Find the acceleration of the mass, the tension in the rope and the speed of the mass having fallen H.

H

Rolling When an object rolls, the point of contact is instantaneous at rest. It can be thought of as an instantaneous axis of rotation, and relative to this point we have pure rotation at that instant. At this instant, each point shown will have a different speed, but is the same for each point on the object.

rolling condition = + Conceptually, we can think of rolling as pure translation at v CM coupled with a simultaneous rotation about CM with

+

We can use energy concepts coupled with kinematics analyze rolling dynamics. H

Exercise: Find and interpret the work done by f s. Exercise: At what angle will slipping start?

Torque A more systematic approach to rotation involves relating the torques associated with forces to the angular accelerations they produce. This can be more complicated if the axis is not fixed, but we shall see that the rolling case is especially simple in this approach. First generalize torque: Right hand rule, RHR Note again that torques are calculated relative to a specific origin and will change if we change the origin. Where could I place an origin above so that F would exert zero torque?

Angular Momentum 1) a vector perpendicular to both and RHR 2) magnitude equals linear momentum times “closest approach distance”. 3) value clearly depends choice of origin. What kind of motion might have constant angular momentum?

Why Angular Momentum? For a system of particles we can define the total angular momentum: Applying the second and third law we get:: Rotational analog of 2 nd law

Dynamics of rolling and slipping We will assume that the axis along which the angular momentum points does not change direction. A baseball curve ball is too hard for us to deal with. Just as we could break the kinetic energy into two parts, so too can we break down the angular momentum: Treat CM as point particle Pure rotation around CM axis

There are two “natural” origins to choose for calculating torques and angular momentum: the center of mass CM and the contact point CP. Let us look at the “L” analysis in each case. The second term will always be CP CM Note the simplicity

Example: C`M

Yo-yo CM

Conservation of Angular Momentum Even if there are external forces present, if they exert no torque about a given axis, then L will be conserved. Note that if L is conserved about one axis, it need not be conserved about any other axis. Disc of mass M rotating with ω o. Bog of mass m lands at center and walks out to rim. Find the final ω

Frictionless table viewed from above. Stick pivoted at one end. Pivot exerts force but no torque about pivot. Linear momentum not conserved, but angular momentum about pivot will be. Was this elastic?

Remove pivot

Axis at bottom as before