Post on 22-Mar-2022
Period T
the number of cycles per second (Hz)
Tf
1
Tf
22
Amplitude A the maximum displacement (m)
the time required to complete one cycle (s)
Frequency f
Angular frequency rad/s
tAAx coscos
Summary of Last Class
HOOKE’S LAW: xkFx
tAvv
sinmax
tAaa
cosmax
2 𝑎 = −𝜔2𝑥 OR
m
k
For a mass m attached to
a spring and set in
vibration on frictionless
horizontal surface
2
21
elasticPE kx joule (J)
10.2 Simple Harmonic Motion and the Reference Circle
Example: 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 sinsinmax
f 2
𝑣𝑚𝑎𝑥 =?
Where?
6283.2 rad/s
1.26 𝑚/𝑠
When passing through
equilibrium, x = 0
𝑣𝑚𝑎𝑥 = 𝐴𝜔
10.3 Energy and Simple Harmonic Motion
Example: Adding a Mass to a Simple Harmonic Oscillator
A 0.20-kg ball is attached to a vertical spring. The spring constant
is 28 N/m. When released from rest, how far does the ball fall
before being brought to a momentary stop by the spring?
𝑥 = 0.07 𝑚
𝑘𝑥 = 𝑚𝑔
10.3 Energy conservation
of EE
2
212
212
212
212
212
21
ooooffff kxmghImvkxmghImv
𝐸 = 𝐾𝐸𝑇 + 𝐾𝐸𝑅 + 𝑃𝐸𝑔 + 𝑃𝐸𝑒
In absence of any external force
Principle of Energy
Conservation
Simple Pendulum
𝜃 𝐿
𝑥
𝑚𝑔
𝜏 = 𝐼𝛼 𝐼 = 𝑚𝑅2
𝐼 = 𝑚𝐿2 𝜏 = −𝑚𝑔𝑥
𝑎𝑇 = 𝐿𝛼 𝛼 =𝑎𝑇𝐿
−𝑚𝑔𝑥 = 𝑚𝐿2𝑎𝑇𝐿
𝑎𝑇 = −𝐴𝜔2 cos 𝜔𝑡 = −𝜔2𝑥
𝜔 =𝑔
𝐿
𝑥
𝜔 = 2𝜋𝑓 𝑓 =𝜔
2𝜋=1
2𝜋
𝑔
𝐿
𝑇 =1
𝑓 𝑇 = 2𝜋
𝐿
𝑔 Period of a simple pendulum is
independent of mass and
Amplitude of the pendulum
As 𝑎𝑇 = 𝑟𝛼
Q: A bob attached to a string and set into simple harmonic (a simple
pendulum is when angle of swing remains small). One such simple pendulum
has a period equals to 0.1 s. Now if the bob is changed to a slightly bigger one
with mass double than the previous bob, keeping length of the string same,
the period of the simple pendulum will
Q: A simple pendulum has a period of 6.0 s on the surface of earth. The
period of the same pendulum on the surface of moon (where the
acceleration due to gravity is 1/6 of that on the surface of earth) will
(c) Remain unchanged (a) Become double (b) Become half
(a) Be the same (b) Increase (c) Decrease
𝑇 = 2𝜋𝐿
𝑔
Problem: A simple pendulum has a ball of mass m attached to a
string of length 1.50 m. The ball is pulled to one side through a small
angle and then released from rest.
(a) After the ball is released how much time is elapsed before it
gains its maximum speed?
(b) After the ball is released how much time is elapsed before it gains
its maximum acceleration?
(c) What will be speed of the ball at the time when the acceleration
is maximum?
(d) What is the angular frequency of this simple pendulum?
(e) What is the linear frequency of this simple pendulum?
Simple pendulum:
0.62 s
1.23 s
0
0.41 Hz
f (Hz) = 1/T 𝜔 =𝑔
𝐿 T = 2π √(L/g)
2.56 rad/s
10.7 Elastic Deformation
STRETCHING, COMPRESSION, AND YOUNG’S MODULUS
AL
LYF
o
Young’s modulus has the units of pressure: N/m2
Table 10.1: Young Modulus of Elasticity for some materials
Q: With same applied force rubber can be elastically deformed
thousands of time more than steel. Does it mean that the Young’s
modulus of elasticity of rubber is
(a) Thousands of time larger than that of steel?
(b) Thousands of time smaller than that of the steel?
(c) Same as that of steel.
10.7 Elastic Deformation
SHEAR DEFORMATION AND THE SHEAR MODULUS
AL
xSF
o
The shear modulus has the units of pressure: N/m2
Deformation is perpendicular to original length
Table 10.2: Shear Modulus of Elasticity for some materials
10.7 Elastic Deformation
Example 14 J-E-L-L-O
You push tangentially across the top
surface with a force of 0.45 N. The
top surface moves a distance of 6.0 mm
relative to the bottom surface. What is
the shear modulus of Jell-O?
AL
xSF
o
xA
FLS o
𝐿0 = .03 𝑚 𝐴 = .07 × .07 𝑚2 ∆𝑥 = .006 𝑚
459 N/m2
10.7 Elastic Deformation
VOLUME DEFORMATION AND THE BULK MODULUS
oV
VBP
The Bulk modulus has the
units of pressure: N/m2
Table 10.3: Bulk Modulus of Elasticity for some materials
10.8 Stress, Strain, and Hooke’s Law
HOOKE’S LAW FOR STRESS AND STRAIN
In general the quantity 𝐹
𝐴 is called the Stress
oLx
N/m2
oLL
SI Unit of Stress:? SI Units of
Strain?
Stress is directly proportional to strain.
oVV
oL
LY
A
F
oL
xS
A
F
oV
VBP
A
FP
The change in the dimension divided by that original is called the Strain
STRETCHING, COMPRESSION SHEAR BULK
Elastic Deformations
A unit less quantity