Rotational Inertia

Circular Motion  Objects in circular motion have kinetic energy. K = ½ m v 2  The velocity can be converted to angular quantities. K = ½ m (r  ) 2 K = ½ (m r 2 )  2 r  m

Integrated Mass  The kinetic energy is due to the kinetic energy of the individual pieces.  The form is similar to linear kinetic energy. K CM = ½ m v 2 K rot = ½ I  2  The term I is the moment of inertia of a particle.

Moment of Inertia Defined  The moment of inertia measures the resistance to a change in rotation. Mass measures resistance to change in velocityMass measures resistance to change in velocity Moment of inertia I = mr 2 for a single massMoment of inertia I = mr 2 for a single mass  The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.

Two Spheres  A spun baton has a moment of inertia due to each separate mass. I = mr 2 + mr 2 = 2mr 2  If it spins around one end, only the far mass counts. I = m(2r) 2 = 4mr 2 m r m

Mass at a Radius  Extended objects can be treated as a sum of small masses.  A straight rod ( M ) is a set of identical masses  m.  The total moment of inertia is  Each mass element contributes  The sum becomes an integral axis length L distance r to r+  r

Rigid Body Rotation  The moments of inertia for many shapes can found by integration. Ring or hollow cylinder: I = MR 2Ring or hollow cylinder: I = MR 2 Solid cylinder: I = (1/2) MR 2Solid cylinder: I = (1/2) MR 2 Hollow sphere: I = (2/3) MR 2Hollow sphere: I = (2/3) MR 2 Solid sphere: I = (2/5) MR 2Solid sphere: I = (2/5) MR 2

Point and Ring  The point mass, ring and hollow cylinder all have the same moment of inertia. I = MR 2I = MR 2  All the mass is equally far away from the axis.  The rod and rectangular plate also have the same moment of inertia. I = (1/3) MR 2  The distribution of mass from the axis is the same. R M M R axis length R M M

Parallel Axis Theorem  Some objects don’t rotate about the axis at the center of mass.  The moment of inertia depends on the distance between axes.  The moment of inertia for a rod about its center of mass: axis M h = R/2

Spinning Energy  How much energy is stored in the spinning earth?  The earth spins about its axis. The moment of inertia for a sphere: I = 2/5 M R 2 The kinetic energy for the earth: K rot = 1/5 M R 2  2 With values: K = 2.56 x J The energy is equivalent to about 10,000 times the solar energy received in one year. next