Motion, Forces and Energy Lectures 10 & 11: Rotational Kinematics When an extended object such as a wheel rotates about its axis, its motion cannot be analysed by treating the object as a particle since, at a given time, different parts of the object have different linear velocities and accelerations. We have to think of a rigid, extended object as a huge number of particles and sum the contributions from each! r s  P x y Angular displacement Average angular speed Angular speed Average angular acceln Angular acceln Remember the relation s=rq and the definition of the radian (arc length / radius).

Rotational Kinematic Equations These equations apply in cases where there is constant angular acceleration. All these rotational equations have the same form as their linear tangential versions.

r s  P x y v Angular and Linear Quantities The CD surface underneath the laser spot moves at a tangential speed of 1.3 ms -1. How far does it travel every time a standard music CD is played?

Rotational Energy riri  mimi x y vivi O Each particle (mass element) has a kinetic energy given by: I = Moment of Inertia

Moment of Inertia The Moment of Inertia is a measure of the resistance of an object to changes in its rotational motion, just as mass is a measure of the tendency of an object to resist changes in its linear motion. Importantly, however, mass is an intrinsic property of the object whereas moment of inertia depends on the geometric arrangement (distribution) of that mass. We use the definition I =  i m i r i 2 but allow each mass element to shrink to zero so that we obtain:

Simple example to illustrate concept M2M2 M1M1 M3M3 M4M4 a a O b b z-axis y x Find moment of inertia about the z-axis I z = M 1 a 2 + M 2 a 2 + M 3 b 2 + M 4 b 2 About the x-axis? I x = M 3 b 2 + M 4 b 2 About the y-axis? I y = M 1 b 2 + M 2 b 2