Physics 218Chapter 15

- Prof. Rupak Mahapatra

Checklist for Today

- Midterm 3 average 61
- Collect your exams from your TAs.
- Last lecture next Monday
- Will cover up to Chpater 18
- Ch 14 and 15 Home work due Wed this week
- Ch 18 Home work due Mon next week

Angular Quantities

- Position ? Angle q
- Velocity ? Angular Velocity w
- Acceleration ? Angular Acceleration a
- Force ? Torque t
- Mass ? Moment of Inertia I
- Today well finish
- Momentum ? Angular Momentum L
- Energy

Rotational Kinetic Energy

- KEtrans ½mv2
- ? KErotate ½Iw2
- Conservation of Energy must take rotational

kinetic energy into account

Rotation and Translation

- Objects can both Rotate and Translate
- Need to add the two
- KEtotal ½ mv2 ½Iw2
- Rolling without slipping is a special case where

you can relate the two - V wr

Rolling Down an Incline

- You take a solid ball of mass m and radius R and

hold it at rest on a plane with height Z. You

then let go and the ball rolls without slipping. - What will be the speed of the ball at the bottom?
- What would be the speed if the ball didnt roll

and there were no friction?

Note Isphere 2/5MR2

Z

A bullet strikes a cylinder

- A bullet of speed V and mass m strikes a solid

cylinder of mass M and inertia I½MR2, at radius

R and sticks. The cylinder is anchored at point 0

and is initially at rest. - What is w of the system after the collision?
- Is energy Conserved?

Rotating Rod

- A rod of mass uniform density, mass m and length

l pivots at a hinge. It has moment of inertia

Iml/3 and starts at rest at a right angle. You

let it go - What is w when it reaches the bottom?
- What is the velocity of the tip at the bottom?

Person on a Disk

- A person with mass m stands on the edge of a disk

with radius R and moment ½MR2. Neither is moving.

- The person then starts moving on the disk with

speed V. - Find the angular velocity of the disk

Same Problem Forces

- Same problem but with Forces

Chapter 18 Periodic Motion

- This time
- Oscillations and vibrations
- Why do we care?
- Equations of motion
- Simplest example Springs
- Simple Harmonic Motion
- Next time
- Energy

Concepts

?The math

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What is an Oscillation?

- The good news is that this is just a fancy term

for stuff you already know. Its an extension of

rotational motion - Stuff that just goes back and forth over and over

again - Stuff that goes around and around
- Anything which is Periodic
- Same as vibration
- No new physics

Examples

- Lots of stuff Vibrates or Oscillates
- Radio Waves
- Guitar Strings
- Atoms
- Clocks, etc
- In some sense, the Moon oscillates around the

Earth

Why do we care?

- Lots of engineering problems are oscillation

problems - Buildings vibrating in the wind
- Motors vibrating when running
- Solids vibrating when struck
- Earthquakes

Whats Next

- First well model oscillations with a mass on a

spring - Youll see why we do this later
- Then well talk about what happens as a function

of time - Then well calculate the equation of motion using

the math

Simplest Example Springs

- What happens if we attach a mass to a spring

sitting on a table at its equilibrium point - (I.e., x 0) and let go?
- What happens if we attach a mass, then stretch

the spring, and then let go?

Questions

- What are the forces?
- Hookes Law F -kx
- Does this equation describe our motion?
- x x0 v0t ½at2

The forces

No force

Force in x direction

Force in x direction

More Detail

Time

Some Terms

Amplitude Max distance Period Time it takes

to get back to here

Overview of the Motion

- It will move back and forth on the table as the

spring stretches and contracts - At the end points its velocity is zero
- At the center its speed is a maximum

Simple Harmonic Motion

- Call this type of motion
- Simple Harmonic Motion
- (Kinda looks like a sine wave)
- Next The equations of motion
- Use SF ma -kx
- (Here comes the math. Its important that you

know how to reproduce what Im going to do next)

Equation of Motion

- A block of mass m is attached to a spring of

constant k on a flat, frictionless surface - What is the equation of motion?

Summary Equation of Motion

- Mass m on a spring with spring constant k
- x A sin(wt f)
- Where
- w2 k/m
- A is the Amplitude
- is the phase
- (phase just allows us to set t0 when we want)

Simple Harmonic Motion

- At some level sinusoidal motion is the definition

of Simple Harmonic Motion - A system that undergoes simple harmonic motion
- is called a
- simple harmonic oscillator

Understanding Phase Initial Conditions

- A block with mass m is attached to the end of a

spring, with spring constant k. The spring is

stretched a distance D and let go at t0 - What is the position of the mass at all times?
- Where does the maximum speed occur?
- What is the maximum speed?

Check

- This looks like a cosine. Makes sense

Spring and Mass Paper which tells us what happens

as a function of time

Example Spring with a Push

- We have a spring system
- Spring constant K
- Mass M
- Initial position X0
- Initial Velocity V0
- Find the position at all times

What is MOST IMPORTANT?

- Simple Harmonic Motion
- X A sin(wt f)
- What is the amplitude?
- What is the phase?
- What is the angular frequency?
- What is the velocity at the end points?
- What is the velocity at the middle?