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.

22212

212

21

iii

iii

iiirot rmrmvmK

i

iirmI 2221 IK rot

2212

21 CMCM IMvK

IfR

vCM (rolling without slipping) and 2kMRI then

)1(221

2

2212

21 kMv

R

vkMRMvK CM

CMCM

i

imM

dmrI 2

dmM

Note: I=kMR2, 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.

22I mL

m m

mm

AxisL

m

m

m

m

Axis

L

2I mL

m m

mmAxis

L

22I mL

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 I1, I2 and I3. Which of the following is true?

A. I1 > I2 > I3 B. I1 > I3 > I2 C. I2 > I1 > I3

3

2

1 I1 = m(2L)2 + m(2L)2 = 8mL2

I2 = mL2 + mL2 + mL2 = 3mL2

I3 = m(2L)2 = 4mL2

L

m

Example: Uniform rod of length L and mass M for rotations about the perpendicular axis through its center.

Density: ML

dx

x

dm

dx

2 3 3 3 32 2 2

2

1 13 2 2 12 12 12

L

L

L L L M LI x dm x dx ML

L

Example: Heavy (real) pulleys. Two blocks of masses m1 and m2 (> m1 ) 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 m1 is initially at rest on the floor and mass m2 hangs at distance h from the floor. Both systems are released simultaneously. In which case does mass m2 hit the floor first?

Case 1; Case 2; Same for both

h

v

v

m2

m1

h

R

2 1

1 2

2

2

m m ghv

Mm m

If no slipping: v = R

24122

412

21

221

MvMRIK

MRI cilinder

2412

2212

121

12 Mvvmvmghmghm

Parallel-axis theorem

22,

2,

2,

2

,2 2 dmrmddrrmdrmrmI icmiicmicmiicmiiii

2MdII cm

EXAMPLE: Rod of mass M and length L about the axis through one end:

Axis A d = L/2

2axis A axis CMI I Md

221

12 2L

ML M

213ML

Axis through CM

2axis CM

112

I ML