simple harmonic motion and elasticitysites.millersville.edu/tgilani/pdf/spring 2018/phys 131...10.1...

Chapter 10

Simple Harmonic Motion and Elasticity

10.1 The Ideal Spring and Simple Harmonic Motion

xkF Appliedx =

spring constant

Units: N/m

𝐹𝐹𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = −𝑘𝑘𝑘𝑘


Example 1 A Tire Pressure Gauge

The spring constant of the springis 320 N/m and the bar indicatorextends 2.0 cm. What force does theair in the tire apply to the spring?

6.4 N


Conceptual Example 2 Are Shorter Springs Stiffer?

A 10-coil spring has a spring constant k. If the spring iscut in half, so there are two 5-coil springs, what is the springconstant of each of the smaller springs?


HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING

The restoring force on an ideal spring is xkFx −=

10.2 Simple Harmonic Motion and the Reference Circle

tAAx ωθ coscos ==

DISPLACEMENT


period T: the time required to complete one cycle

frequency f: the number of cycles per second (measured in Hz)

Tf 1=

Tf ππω 22 ==

amplitude A: the maximum displacement


VELOCITY

tAvvv

Tx ωωθ sinsinmax

−=−=

tAaaa

cx ωωθ coscosmax

2−=−=

ACCELERATION

Radius = A

= −𝜔𝜔2𝑘𝑘


Example 3 The Maximum Speed of a Loudspeaker Diaphragm

The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm. (a)What is the maximum speed of the diaphragm?(b)Where in the motion does this maximum speed occur?

tAvvv

Tx ωωθ sinsinmax

−=−=

fπω 2=3141.5 rad/s

𝑣𝑣𝑚𝑚𝑚𝑚𝑚𝑚 = 0.628 𝑚𝑚/𝑠𝑠

Where?


FREQUENCY OF VIBRATION

mk

=ω

tAax ωω cos2−=tAx ωcos=

xmakx =−

2ωmAkA −=−

For a mass, m attached to a spring and set in vibration on frictionless surface

xmaF =∑kxF −=∑

mkf

π21

=

10.3 Energy and Simple Harmonic Motion

( ) ( ) ( )offo xxkxkxsFW −+== )0cos(cos 21

elasticθ

2212

21

elastic of kxkxW −=


DEFINITION OF ELASTIC POTENTIAL ENERGY

The elastic potential energy is the energy that a springhas by virtue of being stretched or compressed. For anideal spring, the elastic potential energy is

221

elasticPE kx=

SI Unit of Elastic Potential Energy: joule (J)


Example 8 Adding a Mass to a Simple Harmonic Oscillator

A 0.20-kg ball is attached to a vertical spring. The spring constantis 28 N/m. When released from rest, how far does the ball fallbefore being brought to a momentary stop by the spring?

𝑘𝑘𝑘𝑘 = 𝑚𝑚𝑚𝑚

𝑘𝑘 = 0.07 𝑚𝑚

10.3 Energy conservation

of EE =

2212

212

212

212

212

21

ooooffff kxmghImvkxmghImv +++=+++ ωω

𝐸𝐸 = 𝐾𝐾𝐸𝐸𝑇𝑇 + 𝐾𝐾𝐸𝐸𝑅𝑅 + 𝑃𝑃𝐸𝐸𝑆𝑆 + 𝑃𝑃𝐸𝐸𝑒𝑒

In absence of any external force

Simple Pendulum

𝜃𝜃𝐿𝐿

𝑘𝑘

𝑚𝑚𝑚𝑚

�𝜏𝜏 = 𝐼𝐼𝐼𝐼 𝐼𝐼 = 𝑚𝑚𝑅𝑅2

𝐼𝐼 = 𝑚𝑚𝐿𝐿2�𝜏𝜏 = −𝑚𝑚𝑚𝑚𝑘𝑘

𝑎𝑎𝑇𝑇 = 𝐿𝐿𝐼𝐼 𝐼𝐼 =𝑎𝑎𝑇𝑇𝐿𝐿

−𝑚𝑚𝑚𝑚𝑘𝑘 = 𝑚𝑚𝐿𝐿2𝑎𝑎𝑇𝑇𝐿𝐿 𝑎𝑎𝑇𝑇 = −𝐴𝐴𝜔𝜔2 cos 𝜔𝜔𝜔𝜔 = −𝜔𝜔2𝑘𝑘

𝜔𝜔 =𝑚𝑚𝐿𝐿

𝑘𝑘

𝜔𝜔 = 2𝜋𝜋𝜋𝜋 𝜋𝜋 =𝜔𝜔2𝜋𝜋

=12𝜋𝜋

𝑚𝑚𝐿𝐿

𝑇𝑇 =1𝜋𝜋

𝑇𝑇 = 2𝜋𝜋𝐿𝐿𝑚𝑚

Period of a simple pendulum is independent of mass of pendulum

10.7 Elastic Deformation

STRETCHING, COMPRESSION, AND YOUNG’S MODULUS

ALLYFo

∆=

Young’s modulus has the units of pressure: N/m2


SHEAR DEFORMATION AND THE SHEAR MODULUS

AL

xSFo

∆=

The shear modulus has the units of pressure: N/m2


Example 14 J-E-L-L-O

You push tangentially across the topsurface with a force of 0.45 N. The top surface moves a distance of 6.0 mmrelative to the bottom surface. What isthe shear modulus of Jell-O?

AL

xSFo

∆=

xAFLS o

∆=


( )( )( ) ( )

232 mN460

m 100.6m 070.0m 030.0N 45.0

=×

=−

S

xAFLS o

∆=


VOLUME DEFORMATION AND THE BULK MODULUS

∆−=∆

oVVBP

The Bulk modulus has the units of pressure: N/m2

10.8 Stress, Strain, and Hooke’s Law

HOOKE’S LAW FOR STRESS AND STRAIN

Stress is directly proportional to strain.

Strain is a unitless quantity.

SI Unit of Stress: N/m2

In general the quantity F/A is called the stress.

The change in the quantity divided by that quantity is called thestrain:

ooo LxLLVV ∆∆∆

For Practice

Ch 10FOC Questions:

2, 4, 11, 14 and 18.Problems:11, 17, 27, 30, 44, 54, 56 and 92.

simple harmonic motion and elasticitysites.millersville.edu/tgilani/pdf/spring 2018/phys 131...10.1...

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