AP Physics B: Ch.10 - Elasticity and Simple Harmonic Motion

Reading Assignment

Cutnell and Johnson, Physics

Chapter 10

Simple Harmonic motion/systems

Mechanical systems use springs extensivelyImagine a mechanical car engine – lots of moving parts – back and forth movement.

• Prototype for all vibrating systems is simplest possible case called Simple Harmonic Motion (SHM)

Simple Harmonic motion

• Important for understanding many disparate phenomena, e.g., vibration of mechanical structures (bridges, cars, buildings)

• Electrical radio receivers

SpringsObey Hooke’s law in both extension and compression, i.e., F = k x where k is spring constant [N/m]

k is measure of the stiffness of the spring.

In other words the restoring force of the spring depends on its strength and how far it is extended

© John Wiley and Sons, 2004

Springs

Newton’s Third Law => • • The restoring force of an ideal spring is • F = -k x • k is spring constant, • x is displacement from unstrained

length. • If I exert a force to stretch a spring, the

spring must exert an equal but opposite force on me: “restoring force”.

Restoring force is always in opposite direction to displacement of the spring.

Springs

• Stretching an elastic material or spring, and then releasing leads to oscillatory motion. In absence of air resistance or friction we get “Simple Harmonic Motion”

© John Wiley and Sons, 2004

Springs

The motion of the spring in the diagram below is said to be simple harmonic motion.If a pen is attached to an object in simple harmonic motion then the output will be a sine wave.

© John Wiley and Sons, 2004

Springs

Simple harmonic motion is where the acceleration of the moving object is proportional to its displacement from its equilibrium position.It is the oscillation of an object about its equilibrium positionThe motion is periodic and can be described as that of a sine function (or equivalently a cosine function), with constant amplitude. It is characterised by its amplitude, its period and its phase.

Springs

The motion is periodic and can be described as that of a sine function (or equivalently a cosine function), with constant amplitude. It is characterised by its amplitude, its period and its phase.

© John Wiley and Sons, 2004

Springs

• Simple harmonic motion equation –

- is a constant (angular velocity)

xxm

kax

2

© John Wiley and Sons, 2004

Simple Pendulum

Pivot

mg

L

l

s

Tension

•Restoring force acts to pull bob back towards vertical.

•mg sin is the restoring force

•F= mas so restoring force = mas = mg sin

mg cosmg sin

sL

gas

Radius

Arclength

Result:

• Compare with acceleration of mass on a spring.

• Time for one oscillation (T)

Equations are of same form, and we can say for pendulum is equivalent to simple harmonic motion.

sL

gas

xxm

kax

2

L

gpendulum

g

LT

22

Elasticity Experiment

m

Vernier scale

Rigid beam

Metal wire

mg

Resultsmass (kg) Force (N) extension (mm)

0 0 01 9.8 0.142 19.6 0.283 29.4 0.424 39.2 0.565 49 0.76 58.8 0.97 68.6 1.158 78.4 1.459 88.2 2

10 98 2.7

Elasticity Experiment - Results

Elasticity

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100

Force (N)

ex

ten

sio

n (

mm

)

Definitions • Tensile stress =applied force per unit area

• = F/A [N/m2]

• Tensile strain = extension per unit length

• = L/ Lo

• Experiments show that, up to the elastic limit, • Tensile stress Tensile strain (Hooke’s Law) • i.e.,

)"('"constant

xconstant

0

0

YModulussYoungLA

FL

L

L

A

F

Definitions • Young’s Modulus = Stress/Stain

)"('"0 YModulussYoungLA

FL

To measure Young’s Modulus for brass

FYA

LL

L

LY

A

F 0

0

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50

F (N)

exte

nsio

n (m

x 1

0^-3

)

Measure: L0 = 1.00 m A = 0.78 x 10-6 m2

slope = 1.43 x 10-5 m/N

=> Y =

Material Young's modulus (N/m2)

Brass 9.0 x 1010

Aluminium 6.9 x 1010

Steel 2.0 x 1011

Pyrex Glass 6.2 x 1010

Teflon 3.7 x 108

Bone 9.4 x 109 (compression)1.6 x 1010 (tension)

Typical Young’s modulus values

Damped harmonic Motion

In reality, non-conservative forces always dissipate energy in the oscillating system. This is called “damping”.

Lightly damped oscillation

-6

-4

-2

0

2

4

6

0 20 40 60 80 100

time (s)

x (c

m)

Heavily damped oscillation

-4

-2

0

2

4

6

0 20 40 60 80 100

time (s)

x (c

m)

Over-damped motion

0

1

2

3

4

5

6

0 20 40 60 80 100

time (s)

x (c

m)

Resonance

Resonance is the condition where a driving force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. In the absence of damping, resonance occurs when the frequency of the driving force matches the natural frequency of oscillation of the object.

Pushing a swing at the correct time…/incorrect time…resonance

Summary

• Materials subject to elastic deformation obey Hooke’s Law

• Stretching an elastic material or spring, and then releasing leads to oscillatory motion. In absence of air resistance or friction we get “Simple Harmonic Motion”

kxF