Physics 215 -- Fall 2014 Lecture 12-2 1

Welcome back to Physics 215

•Oscillations

•Simple harmonic motion

Physics 215 -- Fall 2014 Lecture 12-2 2

Physics 215 -- Fall 2014 Lecture 12-2 3

Current homework assignment

• HW10:– Knight Textbook Ch.14: 32, 52, 56, 74, 76, 80– Due Wednesday, Nov. 19th in recitation

Physics 215 -- Fall 2014 Lecture 12-2 4

Oscillations

• Restoring force leads to oscillations about stable equilibrium point

• Consider a mass on a spring, or a pendulum

• Oscillatory phenomena also in many other physical systems...

Physics 215 -- Fall 2014 Lecture 12-2 5

Simple Harmonic Oscillator

x0

F=0FF F

Fx = k x

Newton’s 2nd Law for the block:

Spring constant

Differential equation for x(t)

Physics 215 -- Fall 2014 Lecture 12-2 6

Simple Harmonic OscillatorDifferential equation for x(t):

Solution:

Physics 215 -- Fall 2014 Lecture 12-2 7

Simple Harmonic Oscillator

f – frequency Number of oscillations

per unit time T – Period Time taken by one

full oscillation

Units:A - mT - sf - 1/s = Hz (Hertz)

- rad/s

amplitude angular frequency

initial phase

Physics 215 -- Fall 2014 Lecture 12-2 8

Simple Harmonic OscillatorDEMO

• stronger spring (larger k) a faster oscillations (larger f)

• larger mass a slower oscillations

Physics 215 -- Fall 2014 Lecture 12-2 9

Simple Harmonic OscillatorTotal Energy

E =

Physics 215 -- Fall 2014 Lecture 12-2 10

Simple Harmonic Oscillator -- Summary

If F = k x then

Physics 215 -- Fall 2014 Lecture 12-2 11

Importance of Simple Harmonic Oscillations• For all systems near stable equilibrium

– Fnet ~ - x where x is a measure of small deviations from the equilibrium

– All systems exhibit harmonic oscillations near the stable equilibria for small deviations

• Any oscillation can be represented as superposition (sum) of simple harmonic oscillations (via Fourier transformation)

• Many non-mechanical systems exhibit harmonic oscillations (e.g., electronics)

Physics 215 -- Fall 2014 Lecture 12-2 12

(Gravitational) Pendulum Simple Pendulum – Point-like Object

DEMO

0 x

Fnet = mg sin

L

mFor small Fnet is in –x direction:

Fx = mg/L x

mg

T

Fnet

“Pointlike” – size ofthe object small compared to L

Physics 215 -- Fall 2014 Lecture 12-2 13

Two pendula are created with the same length string. One pendulum has a bowling ball attached to the end, while the other has a billiard ball attached. The natural frequency of the billiard ball pendulum is:

1. greater

2. smaller

3. the same

as the natural frequency of the bowling ball pendulum.

Physics 215 -- Fall 2014 Lecture 12-2 14

The bowling ball and billiard ball pendula from the previous slide are now adjusted so that the length of the string on the billiard ball pendulum is shorter than that on the bowling ball pendulum. The natural frequency of the billiard ball pendulum is:

1. greater

2. smaller

3. the same

as the natural frequency of the bowling ball pendulum.

Physics 215 -- Fall 2014 Lecture 12-2 15

(Gravitational) Pendulum Physical Pendulum – Extended Object

DEMO

net = d mg sinFor small :

d m, I

mg

T

sin ≈

net = d mg

Physics 215 -- Fall 2014 Lecture 12-2 16

A pendulum consists of a uniform disk with radius 10cm and mass 500g attached to a uniform rod with length 500mm and mass 270g. What is the period of its oscillations?

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Torsion Pendulum (Angular Simple Harmonic Oscillator)

=

Torsion constant

DEMO

Solution:

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Damped Harmonic Oscillator

For example air drag for small speeds: Fdrag= - b v

Damping constant

Differential equation for x(t)

Solution:

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Forced Harmonic Oscillator

Driving force

Solution yields amplitude of response:

Damping force

Natural frequency

Differential equation for x(t):

Physics 215 -- Fall 2014 Lecture 12-2 20

Resonance

Small (b) damping

Large (b) damping

A

d

Physics 215 -- Fall 2014 Lecture 12-2 21

Reading assignment

• Gravity

• Chapter 13 in textbook