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10.1 The Ideal Spring and Simple Harmonic Motion
xkF Appliedx =
spring constant
Units: N/m
πΉπΉππππππππππππ = βππππ
10.1 The Ideal Spring and Simple Harmonic Motion
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
10.1 The Ideal Spring and Simple Harmonic Motion
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?
10.1 The Ideal Spring and Simple Harmonic Motion
HOOKEβS LAW: RESTORING FORCE OF AN IDEAL SPRING
The restoring force on an ideal spring is xkFx β=
10.2 Simple Harmonic Motion and the Reference Circle
period T: the time required to complete one cycle
frequency f: the number of cycles per second (measured in Hz)
Tf 1=
Tf ΟΟΟ 22 ==
amplitude A: the maximum displacement
10.2 Simple Harmonic Motion and the Reference Circle
VELOCITY
tAvvv
Tx ΟΟΞΈ sinsinmax
β=β=
tAaaa
cx ΟΟΞΈ coscosmax
2β=β=
ACCELERATION
Radius = A
= βππ2ππ
10.2 Simple Harmonic Motion and the Reference Circle
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 ΟΟΞΈ sinsinmax
β=β=
fΟΟ 2=3141.5 rad/s
π£π£ππππππ = 0.628 ππ/π π
Where?
10.2 Simple Harmonic Motion and the Reference Circle
FREQUENCY OF VIBRATION
mk
=Ο
tAax ΟΟ cos2β=tAx Οcos=
xmakx =β
2ΟmAkA β=β
For a mass, m attached to a spring and set in vibration on frictionless surface
xmaF =βkxF β=β
mkf
Ο21
=
10.3 Energy and Simple Harmonic Motion
( ) ( ) ( )offo xxkxkxsFW β+== )0cos(cos 21
elasticΞΈ
2212
21
elastic of kxkxW β=
10.3 Energy and Simple Harmonic Motion
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)
10.3 Energy and Simple Harmonic Motion
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?
ππππ = ππππ
ππ = 0.07 ππ
10.3 Energy conservation
of EE =
2212
212
212
212
212
21
ooooffff kxmghImvkxmghImv +++=+++ ΟΟ
πΈπΈ = πΎπΎπΈπΈππ + πΎπΎπΈπΈπ π + πππΈπΈππ + πππΈπΈππ
In absence of any external force
Simple Pendulum
πππΏπΏ
ππ
ππππ
οΏ½ππ = πΌπΌπΌπΌ πΌπΌ = πππ π 2
πΌπΌ = πππΏπΏ2οΏ½ππ = βππππππ
ππππ = πΏπΏπΌπΌ πΌπΌ =πππππΏπΏ
βππππππ = πππΏπΏ2πππππΏπΏ ππππ = βπ΄π΄ππ2 cos ππππ = βππ2ππ
ππ =πππΏπΏ
ππ
ππ = 2ππππ ππ =ππ2ππ
=12ππ
πππΏπΏ
ππ =1ππ
ππ = 2πππΏπΏππ
Period of a simple pendulum is independent of mass of pendulum
10.7 Elastic Deformation
STRETCHING, COMPRESSION, AND YOUNGβS MODULUS
ALLYFo
β=
Youngβs modulus has the units of pressure: N/m2
10.7 Elastic Deformation
SHEAR DEFORMATION AND THE SHEAR MODULUS
AL
xSFo
β=
The shear modulus has the units of pressure: N/m2
10.7 Elastic Deformation
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
β=
xAFLS o
β=
10.7 Elastic Deformation
( )( )( ) ( )
232 mN460
m 100.6m 070.0m 030.0N 45.0
=Γ
=β
S
xAFLS o
β=
10.7 Elastic Deformation
VOLUME DEFORMATION AND THE BULK MODULUS
ββ=β
oVVBP
The Bulk modulus has the units of pressure: N/m2
10.8 Stress, Strain, and Hookeβs Law
HOOKEβS 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 βββ
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