Rotation of rigid bodies A rigid body is a system where internal forces hold each part in the same relative position

Kinetic Energy and Rotation. Moment of Inertia. If (rolling without slipping) andthen

Note: I=kMR 2, where k 1

Example: Two spheres have the same radius and equal masses. One is made of solid aluminum and the other is a hollow shell of gold. Which one has the biggest moment of inertia about an axis through its center? A. Solid Al B. Hollow Au C. Both the same Hollow gold Solid aluminum Mass is further away from the axis

Example: Moment of inertia of a square of side L made with four identical particles of mass m and four massless rods. m m m m Axis L m m m m L m m m m L The moment of inertia depends on the position and orientation of the axis

Example: Three identical balls are connected with three identical, rigid, massless rods. The moments of inertia about axes 1, 2 and 3 are I 1, I 2 and I 3. Which of the following is true? A. I 1 > I 2 > I 3 B. I 1 > I 3 > I 2 C. I 2 > I 1 > I I 1 = m(2L) 2 + m(2L) 2 = 8mL 2 I 2 = mL 2 + mL 2 + mL 2 = 3mL 2 I 3 = m(2L) 2 = 4mL 2 L m Example: Uniform rod of length L and mass M for rotations about the perpendicular axis through its center. x

Example: Heavy (real) pulleys. Two blocks of masses m 1 and m 2 (> m 1 ) are connected through a string that goes through two different pulleys. In case 1, the pulley is made of plastic. In case 2, the pulley is made of iron. In both cases, mass m 1 is initially at rest on the floor and mass m 2 hangs at distance h from the floor. Both systems are released simultaneously. In which case does mass m 2 hit the floor first? Case 1; Case 2; Same for both h v v  m2m2 m1m1 h R If no slipping: v = R 

Parallel-axis theorem EXAMPLE: Rod of mass M and length L about the axis through one end: Axis A d = L/2 Axis through CM