Physics 215 – Fall 2014 Lecture 11-1 1

Welcome back to Physics 215

Today’s agenda:

• Moment of Inertia

• Rotational Dynamics

• Angular Momentum

Physics 215 – Fall 2014 Lecture 11-1 2

Current homework assignment

• HW8:– Knight Textbook Ch.12: 8, 26, 54, 58, 60, 62, 64– Due Monday, Nov. 5th in recitation

Physics 215 – Fall 2014 Lecture 11-1 3

Discussion

/t) miri2 = net

angular acceleration Moment of inertia, I

I= net

compare this with Newton’s 2nd law

M a = F

Physics 215 – Fall 2014 Lecture 11-1 4

Moment of Inertia* I must be defined with respect to a particular axis

Physics 215 – Fall 2014 Lecture 11-1 5

Moment of Inertia of Continuous Body

r m

m a 0

Physics 215 – Fall 2014 Lecture 11-1 6

Tabulated Results for Moments of Inertia of

some rigid, uniform objects

(from p.299 of University Physics, Young & Freedman)

Physics 215 – Fall 2014 Lecture 11-1 7

Parallel-Axis Theorem

CM

D

D

*Smallest I will always be along axis passing through CM

Physics 215 – Fall 2014 Lecture 11-1 8

Practical Comments on Calculation of Moment of Inertia for Complex Object

1. To find for a complex object, split it into simple geometrical shapes that can be found in Table 9.2

2. Use Table 9.2 to get CMfor each part about the axis parallel to the axis of rotation and going through the center-of-mass

3. If needed use parallel-axis theorem to get for each part about the axis of rotation

4. Add up moments of inertia of all parts

Physics 215 – Fall 2014 Lecture 11-1 9

Beam resting on pivot

M = ? mMb = 2m

NCM of beam

Vertical equilibrium?

Rotational equilibrium?

r r

xrm

F =

=

M =N =

Physics 215 – Fall 2014 Lecture 11-1 10

Suppose M replaced by M/2 ?

• vertical equilibrium?

• rotational dynamics?

• net torque?

• which way rotates?

• initial angular acceleration?

F =

=

Physics 215 – Fall 2014 Lecture 11-1 11

Moment of Inertia?

I = miri2

* depends on pivot position!

I =

* Hence

Physics 215 – Fall 2014 Lecture 11-1 12

Rotational Kinetic Energy

K = i(mivi2 = (1/2)imiri

2

• Hence

K = (1/2)I

• This is the energy that a rigid body possesses by virtue of rotation

Physics 215 – Fall 2014 Lecture 11-1 13

Spinning a cylinderCable wrapped aroundcylinder. Pull off withconstant force F. Supposeunwind a distance d of cable

F

• What is final angular speed of cylinder?

2R

• Use work-KE theorem

W = Fd = Kf = (1/2)I 2

• Mom. of inertia of cyl.? -- from table: (1/2)mR2

– from table: (1/2)mR2

= [2Fd/(mR2/2)]1/2 = [4Fd/(mR2)]1/2

Physics 215 – Fall 2014 Lecture 11-1 14

cylinder+cable problem -- constant acceleration method

extended free body diagramF

FW

N* no torque due to N or FW

* why direction of N ?

* torque due to = FR

* hence = FR/[(1/2)MR2] = 2F/(MR)

t = [4Fd/(MR2)] 1/2

radius R

= (1/2)t2 = d/R; t = [(MR/F)(d/R)]1/2

Physics 215 – Fall 2014 Lecture 11-1 15

Angular Momentum

• can define rotational analog of linear

momentum called angular momentum

• in absence of external torque it will be

conserved in time

• True even in situations where Newton’s laws fail ….

Physics 215 – Fall 2014 Lecture 11-1 16

Definition of Angular Momentum

* Back to slide on rotational dynamics:miri

2/t = i

* Rewrite, using li = miri2

li/t = i

* Summing over all particles in body:

L/t = ext

L = angular momentum = I

pivot

ri

mi

Fi

Physics 215 – Fall 2014 Lecture 11-1 17

An ice skater spins about a vertical axis through her body with her arms held out. As she draws her arms in, her angular velocity

• 1. increases• 2. decreases• 3. remains the same• 4. need more information

Physics 215 – Fall 2014 Lecture 11-1 18

Angular Momentum 1.

p

r

O

Point particle:|L| = |r||p|sin() = m|r||v| sin()

vector form L = r x p – direction of L given by right hand rule

(into paper here)

L = mvr if v is at 900 to r for single particle

Physics 215 – Fall 2014 Lecture 11-1 19

Angular Momentum 2.

o

rigid body:* |L| = I fixed axis of rotation)* direction – along axis – into paper here

Physics 215 – Fall 2014 Lecture 11-1 20

Rotational Dynamics

L/t =

These are equivalent statements

If no net external torque: * L is constant in time * Conservation of Angular Momentum * Internal forces/torques do not contribute to external torque.

Physics 215 – Fall 2014 Lecture 11-1 21

Bicycle wheel demo

• Spin wheel, then step onto platform

• Apply force to tilt axle of wheel

Physics 215 – Fall 2014 Lecture 11-1 22

Linear and rotational motion

• Force

• Acceleration

• Momentum

•

Kinetic energy

• Torque

• Angular acceleration

• Angular momentum

• Kinetic energy

Physics 215 – Fall 2014 Lecture 11-1 23

A hammer is held horizontally and then released. Which way will it fall?

Physics 215 – Fall 2014 Lecture 11-1 24

General motion of extended objects

• Net force acceleration of CM

• Net torque about CM angular acceleration

(rotation) about CM

• Resultant motion is superposition of these two motions

• Total kinetic energy K = KCM + Krot

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1. A.

2. B.

3. C.

4. A and C.

Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown.

After the force has stopped acting on each block, which block will spin the fastest?

Top-view diagram

Physics 215 – Fall 2014 Lecture 11-1 26

1. A.2. B.3. C.4. A, B, and C have the same C.O.M. speed.

Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown.

After each force has stopped acting, which block’s center of mass will have the greatest speed?

Top-view diagram

Physics 215 – Fall 2014 Lecture 11-1 27

Reading assignment

• Rotational dynamics, rolling

• Remainder of Ch. 12 in textbook