1

Class #8

Center of Mass demoMoment of Inertia Moment of Inertia by superposition method

Energy and Conservative forces Force as gradient of potential Line Integrals Curl

Stokes Theorem and Gauss’s Theorem Energy conservation problems – Rolling motion

2

Parallel axis thm worked problemA physicist owns a factory making spherical chocolate Easter eggs with marshmallow centers. The Easter egg has radius “R” and

the marshmallow center has radius “R/3”. a) What is the moment of inertia of a marshmallow filled Easter egg (about an axis through its center of mass)? (Express in terms of

M, the mass of a SOLID chocolate sphere of radius R)  b) If the machine malfunctions and offsets the marshmallow center by a distance of R/3 from the axis of rotation through the original

center of the object, what is the new moment of inertia of the entire defective Easter egg about the center of the sphere?  

3

L8-1 Disk with Hole

Axis 1 (through CM)Axis 2

(Parallel to axis 1)

d

R, M

The solid disk of radius R initially has mass M. A hole is cut in it as indicated. What is moment of inertia of disk with hole spun about axis 1?

r

Given:r=R/3d=R/2M=mass of the diskw/o a hole in it.Density of disk isUniform.

4

Kinetic energy is useful because it relates velocity change to work done.

Theorem is valid even if force is not conservative.

Work – Kinetic Energy Theorem

0 0 0

0

0

2

1 21

( )

( )

2 20 0

( )

1 1( ) ( )

2 2

P

P PP

r r v r

r r v r

v

v

r r

F dr Work

dv drF r dr m dr m dv

dt dt

mv dv mv r mv r

Work KE

5

A force is conservative iff:

1. The force depends only on

2. For any two points P1, P2 the work done by the force is independent of the path taken between P1 and P2.

1. Equivalent to

Conservative Forces

r( )

( , , )

F F r

F F r r t

, .Dependence on r t not allowed

0

2

1

2

1

0

( )

( 1 2)

( ) ( ) ( )

P

P

P

P

r

r

F dr Const

over all paths

F dr Work P P

U r W r r F r dr

6

Force as the gradient of potential

00( ) ( ) ( ) ( )

( ) ( )

x r

r

r rx x

Scalar Vector

F x f x dx U r F r dr

dFf x F r U

dx

ˆ ˆ ˆ

1 1ˆ ˆˆsin

x y zx y z

rr r r

3 sinU Axy B Cz

7

Gravitational Potential

ˆ ˆ ˆ

1 1ˆ ˆˆsin

x y zx y z

rr r r

2

( )

1

1

0

r

GMmU r

r

F U GMmr

GMmF GMm

r r r

F F

8

Line integral and Closed loop integral

Conservative force

a) P1 and P2 with two possible integration paths. b) and c) P1 and P2 are brought closer together. d) P1 and P2 brought together to an arbitrarily small distance . Geometric

argument that conservative force implies zero closed-loop path integral.

1

20

0 0

P

PF dr Const F dr

F dr F

9

Integration by eyeball

Conservative force

0F dr

a

c d

b

e f

10

Curl as limit of tiny line-integrals

0ˆ( ) lim

ia i

F drcurl F n

a

11

Stokes and Gauss’s theorem’s

Gauss – Integrating divergence over a volume is equivalent to integrating function over a surface enclosing that volume.

Stokes – Integrating curl over an area is equivalent to integrating function around a path enclosing that area.

12

The curl-o-meter (by Ronco®)

Conservative force

0 0F dr F

a

c d

b

e fˆ ˆ ˆ

det

x y z

x y z

fx y z

f f f

13

L8-2 – Area integral of curl

ˆ ˆ( )F r yx xy

O

y

xP(1,0)

Q(0,1)

a

c

b

Calculate, along a,c

Calculate, along a,b

Calculate, inside a,c

Calculate, inside a,b

OQPF dr

OQP

F dA

14

L8-3 Energy problems

A block of mass “m” starts from rest and slides down a ramp of

height “h” and angle “theta”. Use energy conservation

methods to:

a) Calculate velocity “v” at bottom of ramp

b) Do part “a” again, in the presence of friction “”

c) Do the same for a rolling disk (mass “m”, radius r)

O

y

x

m

h

m

y

x