Ch.10 Elasticity & Oscillations

• Problems: 3, 4, 27, 29.

• Elastic deformation

• Hooke’s Law

• Simple Harmonic Motion (SHM)

• period & frequency of SHM

• (sections 3, 4, 7, 8, 9, 10)

Hooke’s Law

• The restoring force exerted by an object is proportional to the amount the object has been deformed.

• F = -kx

• k = elastic constant (N/m)

• Ex. SHM-springs used in lab are about 8 N/m.

• /

Young’s Modulus: Object with length = L, and area = A.

L

LY

A

F

Y is Young’s Modulus which is a measure of stiffness.

k = YA/L.

Ex. Young’s Modulus. Area = 0.1m^2, L = 25m, Y = 5000 N/m^2. A pulling force of 100N applied at each end will cause the length to change by:

m 5N/m 5000

m 25

m 1.0

N 10022

Y

L

A

FL

Ex. An object with Area = 0.0001m^2, L = 1m, is pulled on both ends by a force of 10N, causing its length to increase to 1.4m. Young’s modulus is:

22

/000,2504.0

m 1

m 0001.0

N 10mN

m

L

L

A

FY

6

vibrations

• Examples:

• vibrating reed, mass on spring,

• drum, piano wire, string,…

• most vibrations are sinusoidal in time,

• and called “simple harmonic” motions (shm)

7

terminology• x: displacement

• A = amplitude = maximum displacement

• f: frequency (cycles/s) T = 1/f

• angular frequency f: (rad/s)

• k: spring constant (N/m)

8

sinusoidal nature of vibrations

m

kf

2

1

A 0.10kg mass is hung from a spring with spring constant 8N/m.

Also calculate for 0.2kg

mk

mgx 123.0

8

)8.9)(10.0(

Hzm

kf 43.1

1.0

8

2

1

2

1

Hzm

kf 01.1

2.0

8

2

1

2

1

10

SHM Formulas)2cos( ftAx

fAAv 2max AfAa 222max 4

m

kf

2

1

2max2

12212

212

21 mvkAmvkxE

Position:

frequency:

maximums:

11

SHM Formulas• position:

• frequency:

• maximums:

)2cos( ftAx

fAAv 2max AfAa 222max 4

m

kf

2

1

2max2

12212

212

21 mvkAmvkxE

Ex: SHM

2max2

12212

212

21 mvkAmvkxE

Ex: k = 10N/m, m = 200grams, A = 10cm.

JkAE 05.0)1.0)(10( 2212

21

Summary

•Young’s Modulus

•Hooke’s Law

•Simple Harmonic Motion

•SHM Examples: Mass-Spring System,

•Energy Conservation Applied to SHM

14

natural frequency

• lowest frequency an object vibrates with when struck

• also called “resonant frequency”

• Demo: Driving Tuning Fork

• /

15

resonance

• absorb max energy: frequency = resonant frequency

• example: 256 Hz guitar string resonates when exposed to 256 Hz.

• Chilandi plates video

16

Main ResultskxF

Av max

)cos( tAx T

2

2max Aa

m

k

2

f

221 kxPEelastic

2212

21 kxmvEmech

17

sinusoidal nature of shm

• position of blue mass moving on spring turns out to be same as the horizontal position of an object in uniform circular motion.

)cos( tAx

18

L

gf 2 small angles

Simple Pendulum