10.1 The Ideal Spring and Simple Harmonic Motion
xkF Appliedx =
spring constant
Units: N/m
๐น๐น๐๐๐๐๐๐๐๐๐๐๐๐ = โ๐๐๐๐
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
HOOKEโS LAW: RESTORING FORCE OF AN IDEAL SPRING
The restoring force on an ideal spring is xkFx โ=
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 ฯฯฯ 22 ==
amplitude A: the maximum displacement
10.2 Simple Harmonic Motion and the Reference Circle
VELOCITY
tAvvv
Tx ฯฯฮธ sinsinmax
โ=โ=
tAaaa
cx ฯฯฮธ coscosmax
2โ=โ=
ACCELERATION
Radius = A
= โ๐๐2๐๐
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 ฯฯฮธ sinsinmax
โ=โ=
fฯฯ 2=3141.5 rad/s
๐ฃ๐ฃ๐๐๐๐๐๐ = 0.628 ๐๐/๐ ๐
Where?
10.2 Simple Harmonic Motion and the Reference Circle
FREQUENCY OF VIBRATION
mk
=ฯ
tAax ฯฯ cos2โ=tAx ฯcos=
xmakx =โ
2ฯmAkA โ=โ
For a mass, m attached to a spring and set in vibration on frictionless surface
xmaF =โkxF โ=โ
mkf
ฯ21
=
10.3 Energy and Simple Harmonic Motion
( ) ( ) ( )offo xxkxkxsFW โ+== )0cos(cos 21
elasticฮธ
2212
21
elastic of kxkxW โ=
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?
๐๐๐๐ = ๐๐๐๐
๐๐ = 0.07 ๐๐
10.3 Energy conservation
of EE =
2212
212
212
212
212
21
ooooffff kxmghImvkxmghImv +++=+++ ฯฯ
๐ธ๐ธ = ๐พ๐พ๐ธ๐ธ๐๐ + ๐พ๐พ๐ธ๐ธ๐ ๐ + ๐๐๐ธ๐ธ๐๐ + ๐๐๐ธ๐ธ๐๐
In absence of any external force
Simple Pendulum
๐๐๐ฟ๐ฟ
๐๐
๐๐๐๐
๏ฟฝ๐๐ = ๐ผ๐ผ๐ผ๐ผ ๐ผ๐ผ = ๐๐๐ ๐ 2
๐ผ๐ผ = ๐๐๐ฟ๐ฟ2๏ฟฝ๐๐ = โ๐๐๐๐๐๐
๐๐๐๐ = ๐ฟ๐ฟ๐ผ๐ผ ๐ผ๐ผ =๐๐๐๐๐ฟ๐ฟ
โ๐๐๐๐๐๐ = ๐๐๐ฟ๐ฟ2๐๐๐๐๐ฟ๐ฟ ๐๐๐๐ = โ๐ด๐ด๐๐2 cos ๐๐๐๐ = โ๐๐2๐๐
๐๐ =๐๐๐ฟ๐ฟ
๐๐
๐๐ = 2๐๐๐๐ ๐๐ =๐๐2๐๐
=12๐๐
๐๐๐ฟ๐ฟ
๐๐ =1๐๐
๐๐ = 2๐๐๐ฟ๐ฟ๐๐
Period of a simple pendulum is independent of mass of pendulum
10.7 Elastic Deformation
STRETCHING, COMPRESSION, AND YOUNGโS MODULUS
ALLYFo
โ=
Youngโs modulus has the units of pressure: N/m2
10.7 Elastic Deformation
SHEAR DEFORMATION AND THE SHEAR MODULUS
AL
xSFo
โ=
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
โ=
xAFLS o
โ=
10.7 Elastic Deformation
( )( )( ) ( )
232 mN460
m 100.6m 070.0m 030.0N 45.0
=ร
=โ
S
xAFLS o
โ=
10.7 Elastic Deformation
VOLUME DEFORMATION AND THE BULK MODULUS
โโ=โ
oVVBP
The Bulk modulus has the units of pressure: N/m2
10.8 Stress, Strain, and Hookeโs Law
HOOKEโS 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 โโโ