Chap. 14: Oscillations, Periodic Motion, Simple Harmonic Motion 1 Characterized by: Period (T) and Frequency (f) Preparation for: Mechanical wave motion Electromagnetic wave Dynamics: F = m a, t = I a Equation of Motion: General solution:
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Chap. 14: Oscillations,
Periodic Motion,
Simple Harmonic Motion
1
Characterized by:
Period (T) and Frequency (f)
Preparation for:
Mechanical wave motion
Electromagnetic wave
Dynamics: F = m a, t = I a
Equation of Motion:
General solution:
Looking Back to Chap. 7 • To describe oscillations in terms of amplitude, period, frequency and angular frequency
• To do calculations with simple harmonic motion (SHM); To analyze simple harmonic motion using energy
• To apply the ideas of simple harmonic motion to different physical situations
• To analyze the motion of a simple pendulum, followed by a physical pendulum
• To explore how oscillations die out • To learn how a driving force can cause
resonance
2
Oscillations
What is S.H.M.?
5
Periodic Motion Horizontal oscillation Vertical oscillation Vibration
Common characteristics Simplified Model Simple Harmonic Motion
Oscillations
S.H.M.
Useful Math and Physics
Trig. functions:
sin(q + p/2) = cos(q)
sin(q + p ) = -sin(q)
cos(q - p/2) = sin(q)
Derivative and integral
Trig. functions
Approximation:
sinq ~ q
S.H.O.
1) Spring plus block
Horizontal
Vertical
2) Pendulum
Simple pendulum
Physical pendulum
F = m a
t = I a
6
Oscillations
Circular Motion to SHM
x
R0 q
Starts here
7
Oscillations
Analyzing Spring+Block System
9
Oscillations
S.H.M. : Spring+Block system and
Simple Pendulum system
Spring+block System
Simple Pendulum
11
Oscillations
Physical Pendulum – S.H.M.?
13
Oscillations
Physical Pendulum – S.H.M.?
Physical Pendulum
Simple Pendulum
m
Axis of rotation
14
Oscillations
Math about a and a
Oscillations
Physical Pendulum Physical Pendulum –– S.H.M.?S.H.M.?
Physical PendulumPhysical Pendulum
Simple PendulumSimple Pendulum
Oscillations
S.H.M. : Spring+Block system and S.H.M. : Spring+Block system and
Simple Pendulum systemSimple Pendulum system
Spring+block SystemSpring+block System
Simple PendulumSimple Pendulum
15
Oscillations
S.H.M. – Dynamics (Summary)
xCtd
xd
xC
DtCAC
DtCACCtd
xd
DtCACdt
dx
DtCAtx
Ctd
dxC
td
xd
)cos(
)cos(
)sin(
:then ),cos()( If
or
2
2
2
2
2
2
2
2
-
-
-
-
-
-- q
q
Math about C Physics about C
)cos ()( φω tAtx 18
Oscillations
Circular Motion to SHM
x
x = R0 cos q
where q = w t
R0
x(t) = R0 cos (w t)
q
Starts here
Kin. Equation of SHM
x(t) = R0 cos ( )
f
19
Oscillations
Acceleration is proportional to displacement.
S.H.M.
21
XCtd
Xd
2
2
-
Oscillations
What is f ?
23
t
x(t) = A sin( w t )
= A cos( w t )
= A cos( w t - f/w) )
= A cos( w t - f )
x(t)
f/w
Oscillations
What is f ?
25
x(t) = A sin( w t ) = A cos[ w t + (- p/2) ]
= A cos[ w t - p/2w) ]
= A cos[ w t - 2p/4w) ]
t (2p/4) w = (2p/w)/4 = T/4
Oscillations
S.H.M.
3 independent variables:
Angular frequency
Amplitude
Phase angle
x(t) = A cos(w t + f)
q(t) = A cos(w t + f)
Period:
T = 1 / f T = 2p/w
Note: Angular frequency (w)
is NOT the angular velocity.
Example of x(t),
where f = 0
Quick Check:
How do v(t) and a(t) look like?
26
Oscillations
Graphs
27
Oscillations
Technical Steps
Oscillations
S.H.M.S.H.M.
3 independent variables:
Angular frequency
Amplitude
Phase angle
x(t) = A cos(w t + f)
q(t) = A cos(w t + f)
Period:
T = 1 / f T = 2p/w
Period:
T = 1 / f T = 2p/w
Note: Angular frequency (w)
is NOT the angular velocity.
Example of x(t),
where f = 0
Quick Check:Quick Check:
How do v(t) and a(t) look like?
29
Oscillations 31
Oscillations 32
Oscillations 33
An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object’s maximum speed vmax?
A. T and vmax both double.
B. T remains the same and vmax doubles.
C. T and vmax both remain the same.
D. T doubles and vmax remains the same.
E. T remains the same and vmax increases by a factor of
Q14.1
2.
This is an x-t graph for an object in simple harmonic motion.
A. t = T/4
B. t = T/2
C. t = 3T/4
D. t = T
Q14.2
At which of the following times does the object have the most negative velocity vx?
This is an x-t graph for an object in simple harmonic motion.
A. t = T/4
B. t = T/2
C. t = 3T/4
D. t = T
Q14.3
At which of the following times does the object have the most negative acceleration ax?
This is an ax-t graph for an object in simple harmonic motion.
A. t = 0.10 s
B. t = 0.15 s
C. t = 0.20 s
D. t = 0.25 s
Q14.4
At which of the following times does the object have the most negative displacement x?
This is an ax-t graph for an object in simple harmonic motion.
A. t = 0.10 s
B. t = 0.15 s
C. t = 0.20 s
D. t = 0.25 s
Q14.5
At which of the following times does the object have the most negative velocity vx?
A. t = T/8
B. t = T/4
C. t = 3T/8
D. t = T/2
E. more than one of the above
This is an x-t graph for an object connected to a spring and moving in simple harmonic motion.
Q14.6
At which of the following times is the potential energy of the spring the greatest?
A. t = T/8
B. t = T/4
C. t = 3T/8
D. t = T/2
E. more than one of the above
This is an x-t graph for an object connected to a spring and moving in simple harmonic motion.
Q14.7
At which of the following times is the kinetic energy of the object the greatest?
To double the total energy of a mass-spring system oscillating in simple harmonic motion, the amplitude must increase by a factor of
A. 4.
B.
C. 2.
D.
E.
Q14.8
2 1.414.
4 2 1.189.
2 2 2.828.
A simple pendulum consists of a point mass suspended by a massless, unstretchable string.
If the mass is doubled while the length of the string remains the same, the period of the pendulum
A. becomes 4 times greater.
B. becomes twice as great.
C. becomes greater by a factor of .
D. remains unchanged.
E. decreases.
Q14.9
2
Oscillations
Practice Problems
52
Oscillations
(Challenging) Practice Problems
53
Oscillations
Rotational Motion
2(e) A solid disk (mass M = 3.00 kg and radius R = 20.0 cm) is
hung from the wall by means of a metal pin through the hole,
and used as a pendulum. Calculate the moment of inertia of the
disk about the pin (= the axis of the rotation).
Rotational Motion
2(c) A meter stick (mass M = 0.500 kg and length L = 1.00 m) is
hung from the wall by means of a metal pin through the hole,
and used as a pendulum. Express the moment of inertia of the
stick about the pin (= the axis of the rotation) in terms of M, L,
and x.
Rotational Motion
Torque due to Gravity?Torque due to Gravity?
)(sin2
-
q
t
mg xl
F r
r
x
F
lCM
mass m
? F r
t
Looking Back at I and t
54
Oscillations
Concepts of S.H.M. Dynamics & Kinematics:
Spring+block
F = m a & (x, v, a)
Pendulum
a part of circular motion t = I a & (q, w, a)
Force: Conservative force
Restoring force
Conservation: K + U = constant
+ S.H.M. (w as angular frequency) 55
Oscillations
Work Sheet 1
Name: _______________ Section: _______ SID: __________
Work on (a), (b), (c), and (d)
56
Oscillations
Equation of Motion:
d2x/dt2 + w2 x = 0 or d2q/dt2 + w2 q = 0
General solution:
x(t) = A cos(w t + f)
q(t) = A cos(w t + f)
Position (x or q) as a function of time.
Amplitude
Angular Frequency
Oscillations
Periodic Motion
Simple Harmonic Motion
(S.H.M.)
57
Oscillations
Example 7: Angular Velocity?
58
Oscillations
How to find T?
Find I
Calculate w
Then, find T
Want to find T
Need to know w
Then, find I
59
Oscillations
Example 6: Physical Pendulums (1)
Exercise 6.1: Express w
= [Find I]
60
Oscillations
Example 6: Physical Pendulums (2)
Exercise 6.2: Express a = [Find I and d]
61
Oscillations
Example 6: Physical Pendulums (3)
Exercise 6.3: Express w
= [Find I and d] Steps: 1. What is asked?
w (rigid body) I and d
2. How to find I and d for a system of two rigid bodies?
I = I1 + I2
3. How to find I1 (or I2)?
Where is the c.m. position? Use parallel-axis theorem
4. How to find d ?
Where is the c.m. position of the system?
I
mgd
I
mgd- wqa
62
Oscillations
Example 6: Physical Pendulums (3)
63
Oscillations
Problem 4: (25 points)
Determine the net torque (magnitude and direction) due to gravity on the system about the
pin, shown in the figure below. A beam has mass M and length l; a big solid sphere has
mass M and radius R; a small sphere has mass M/2 and radius R/2. Assume l > R and l > 2x.
Also determine the moment of inertia of the system about the pin. [Hint: use Parallel-axis
theorem.]
l
But, this can also be a Chap.14 problem,
If I ask you to find w. (angular frequency)
in S.H.M. of a physical pendulum.
Chap. 10
64
Oscillations
Work Sheet 3
Name: _______________ Section: _______ SID: __________
Work on (a)
Repeat (a) without and with a particle (mass m)
65
Oscillations
Example 8: Physical Pendulums
2.00 cm
66
Oscillations 67
Oscillations
x = 2.00 cm
68
Oscillations
Periodic Motion
T = 2p L cosq / g T 2 = (4p2/GM) s 3
69
Oscillations
Example 1: (a) FT = ? (b) T = ?
mg sinq
Be Critcal Thinker
g
LT
qp
cos2
g
lT p2
FT=mgcosq FT=mg/cosq
70
Oscillations
S.H.M. : T = 2p/w
72
Oscillations
Example 2(A)
+
+ +
+ + 0
T/4
T/2
3T/4
T
Equilibrium
Positions
A
73
Oscillations
Example 2(B)
Ea = Eb = Ec = Ed
74
Oscillations
Example 4: Momentum conservation!
What is the speed of the bullet?
Express the speed in terms of m, M, k, and d.
d
76
Oscillations
Recap from the Previous Lecture
77
Oscillations
Example 5
You hold the block
at x = A (= 0.030 m)
by applying 6.0 N.
Then, the block was
released.
The motion of the block undergoes SHM.
Can you show that a = –(k/m) x ?
Also find: (a) k
(b) w
(c) T
(d) vmax (where?, when?)
(e) x, v and a at t = 2 sec
k 0.50 kg
78
Oscillations
Example 9: Vertical S.H.M.
79
Oscillations
Physical Pendulum (I)
q
q
Small oscillation
q = small
L|| = L cosq ~ L
L||
80
Oscillations
Physical Pendulum (II)
+
x
q
q
+
mg
x0
x = 0
Fsp = k x0
Equilibrium (Sti = 0)
mg(L/2) – kx0(L) = 0
81
Oscillations
Physical Pendulum (III)
+
x
q
q
+
mg
x0
x = 0 Fsp = k x
Equation of motion (Sti = I a)
Sti = mg(L/2) – kx(L) = kx0(L) – kx(L)
∴ – k(x–x0)(L) = Irod(P) d2q/dt2
P
+ x
Lq = x – x0
82
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