#### Transcript Rotational Inertia

Rotational Inertia Circular Motion Objects in circular motion have kinetic energy. • K = ½ m v2 v r w m The velocity can be converted to angular quantities. • K = ½ m (r w)2 • K = ½ (m r2) w2 Integrated Mass Ki 1 2 ( m i )( w ri ) Ki 1 2 w ( ri m i ) K rot 1 2 The kinetic energy is due to the kinetic energy of the individual pieces. The form is similar to linear kinetic energy. 2 2 2 K rot 1 2 Iw w ( ri m i ) w 2 2 2 1 2 2 r i 2 m • KCM = ½ m v2 • Krot = ½ I w2 i 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 velocity • Moment of inertia I = mr2 for a single mass The total moment of inertia is due to the sum of masses at a distance from the axis of rotation. N I i 1 m i ri 2 Two Spheres w A spun baton has a moment of inertia due to each separate mass. • I = mr2 + mr2 = 2mr2 m r m If it spins around one end, only the far mass counts. • I = m(2r)2 = 4mr2 Mass at a Radius Extended objects can be treated as a sum of small masses. I The total moment of inertia is A straight rod (M) is a set of identical masses Dm. ( m i ) ri 2 Each mass element contributes mi (M / L )Dr I ( M / L ) ri D r 2 distance r to r+Dr length L The sum becomes an integral I (1 / 3 ) ML axis 2 Rigid Body Rotation The moments of inertia for many shapes can found by integration. • Ring or hollow cylinder: I = MR2 • Solid cylinder: I = (1/2) MR2 • Hollow sphere: I = (2/3) MR2 • Solid sphere: I = (2/5) MR2 Point and Ring The point mass, ring and hollow cylinder all have the same moment of inertia. • I = MR2 The rod and rectangular plate also have the same moment of inertia. • I = (1/3) MR2 All the mass is equally far away from the axis. The distribution of mass from the axis is the same. M R M R M M length R axis length R Parallel Axis Theorem Some objects don’t rotate about the axis at the center of mass. The moment of inertia for a rod about its center of mass: h = R/2 M The moment of inertia depends on the distance between axes. I I CM Mh 2 axis (1 / 3 ) MR 2 I CM M ( R / 2 ) I CM (1 / 3 ) MR 2 I CM (1 / 12 ) MR (1 / 4 ) MR 2 2 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 R2 • The kinetic energy for the earth: Krot = 1/5 M R2 w2 The energy is equivalent to about 10,000 times the solar energy received in one year. • With values: K = 2.56 x 1029 J next