Physics 218, Lecture XVII 1

Physics 218Lecture 17 Dr. David Toback

Physics 218, Lecture XVII 2

Before we begin

• Chapter 8HW was due Monday• Ch. 10 reading assignment: due

today• Already covered most of chapter

9 in lecture, but I’m changing the Chapter 9 HW due date to next Wednesday

• Ch. 10HW still due the following Monday

Physics 218, Lecture XVII 3

Rotational Motion

Chapters 9 and 10 in four combined lectures

• This is the 2nd of the 4 lectures

• Concentrate on the relationship between linear and angular variables

• Last time started kinematics… Move to dynamics just like earlier this semester

Physics 218, Lecture XVII 4

Angular QuantitiesLast time:• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration This time we’ll start by discussing the

vector nature of the variables and then move forward on the others:– Force– Mass– Momentum– Energy

Physics 218, Lecture XVII 5

Physics 218, Lecture XVII 6

Angular Velocity and Acceleration

Are and vectors?

and clearly have magnitude

Do they have direction?

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Right-Hand Rule

Yes!Define the direction to point along the axis of rotation

Right-hand Rule

This is true for and

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Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Moving forward:

– Force

– Mass

– Momentum

– Energy

Physics 218, Lecture XVII 9

Torque

• Torque is the analogue of Force

• Take into account the perpendicular distance from axis– Same force further from the axis leads to more

Torque

Physics 218, Lecture XVII 10

Slamming a door

We know this from experience:–If we want to slam a door really

hard, we grab it at the end

–If we try to push in the middle, we aren’t able to make it slam nearly as hard

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Torque Continued

• What if we change the angle at which the Force is applied?

• What is the “Effective Radius?”

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Slamming a door

We also know this from experience:– If we want to slam a door really

hard, we grab it at the end and “throw” perpendicular to the hinges

– If we try to pushing towards the hinges, the door won’t even close

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Torque

• Torque is our “slamming” ability

• Write Torque as

• To find the direction of the torque, wrap your fingers in the direction the torque makes the object twist

Fr

Fr

sin||||||

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Torque and Force

Torque problems are like Force problems

1. Draw a force diagram2. Then, sum up all the torques to

find the total torque

Is torque a vector?

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Example: Composite Wheel

Two forces, F1 and F2, act on different radii of a wheel, R1 and R2, at different angles 1 and 2. 1is a right angle.

If the axis is fixed, what is the net torque on the wheel? F1

F2

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Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration Moving forward:

– Force Torque – Mass

– Momentum

– Energy

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Analogue of Mass

The analogue of Mass is called Moment of Inertia

Example: A ball of mass m moving in a circle of radius R around a point has a moment of inertia

F=ma =

Physics 218, Lecture XVII 18

Calculate Moment of Inertia

Calculate the moment of inertia for a ball of mass m relative to the center of the circle R

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Moment of Inertia

dmrImrI 22 or

• To find the mass of an object, just add up all the little pieces of mass

To find the moment of inertia around a point, just add up all the little moments

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Torque and Moment of Inertia

• Force vs. Torque

F=ma = I

• Mass vs. Moment of Inertia

dmrI

mrIm

2

2

or

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Pulley and Bucket

A heavy pulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The friction torque is fric.

Find the angular acceleration

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Spherical Heavy Pulley

A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force FT. It accelerates to an angular speed of in time t.

What is the moment of inertia of the pulley?

R

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Less Spherical Heavy PulleyA heavy pulley, with radius R,

starts at rest. We pull on an attached rope with constant force FT. It accelerates to final angular speed in time t.

A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this fric.

What is this better estimate of the moment of Inertia?

R

Physics 218, Lecture XVII 24

Next Time

• The rest of Chapter 10– More on angular “Stuff”

– Angular Momentum

– Energy

• Get caught up on your homework!!!

• Chapter 8 was due Monday, Chapter 9 homework due Wednesday, Chapter 10 is due the Monday after that

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Example of Cross Product

The location of a body is length r from the origin and at an angle from the x-axis. A force F acts on the body purely in the y direction.

What is the Torque on the body?

z

x

y

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Calculate Moment of Inertia

1. Calculate the moment of inertia for a ball of mass m relative to the center of the circle R

2. What about lots of points? For example a wheel

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Rotating RodA uniform rod of mass m, length l, and moment of

inertia I = ml2/3 rotates around a pivot. It is held horizontally and released.

Find the angular acceleration and the linear acceleration a at the end. Where, along the rod, is a = g?

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Two weights on a bar

Find the moment of inertia for the two different Axes

middle

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Schedule Changes

Please see the handout for schedule changes

New Exam 3 Date:Exam 3

Tuesday Nov. 26th

Physics 218, Lecture XVII 31

Moments of Inertia