15: Chapter Oscillatory motion
Reading assignment:
Chapter 15.1 to 15.5; 15.6 & 15.7 cursory
Homework : QQ1, QQ2, QQ3, QQ4, QQ6, OQ1, OQ3, OQ, 4, OQ5, OQ7, OQ9, OQ10,
AE1, AE6, 4, 5, 8, 9, 15, 18, 19, 27, 29, 37
Due date: Tuesday, April 26
Well deal mainly with simple harmonic
oscillations where the position of the object
is specified by a sinusoidal function.
Mass on a spring
Energy
Pendulum
Simple harmonic motion/oscillation
-
Block attached to a spring
-
Motion of a swing
-
Motion of a pendulum (mathematical, physical)
-
Vibrations of a stringed musical instrument
-
Motion of a cantilever
-
Oscillations of houses, bridges,
-
All clocks use simple harmonic motion
Piezoelectric (quartz) tuning fork from a wrist watch
(In a piezoelectric material, distortion creates voltage and vice versa)
Brief Aside: The Importance of Time & The Longitude Problem
1)
2)
Harrisons H4
1759
Harrisons H1 1735
http://www.rog.nmm.ac.uk/museum/harrison/longprob.html
3)
Determine local time through
sun.
Compare with time (accurate
clock!) at port of departure
(London).
Each hour difference
corresponds to 15o longitude
(360o/24 hours).
Simple harmonic motion/oscillation
Restoring force:
F = - kx
Acceleration:
k
a = x
m
Acceleration and restoring force:
-
proportional to x
-
directed toward the equilibrium position
Simple harmonic motion/oscillation
An object moves with simple harmonic motion whenever
its acceleration is proportional to its displacement from
some equilibrium position and is oppositely directed.
a = 2 x
d 2x
2
= x
2
dt
Solution to this 2nd order differential equation is:
x(t ) = A cos(t + )
or:
x(t ) = A sin(t + )
Properties of simple harmonic motion
Position of particle at time t:
x(t ) = A cos(t + )
Aamplitude
Angular frequency
phase constant, phase angle
Tperiod
(t + ) phase
x(t ) = A cos(t )
Properties of simple harmonic motion
The period T of the motion is the time it takes for the particle to complete one full cycle
Periodic motion
The value of x at time t is equal to the value of x at time t +T
Properties of simple harmonic motion
Displacement:
Period T:
2
T=
x(t ) = A cos(t + )
Angular frequency:
Frequency:
1
f= =
T 2
2
= 2f =
T
Units: 1/s = 1 Hz
Velocity:
v(t ) = A sin(t + )
Acceleration:
a (t ) = A cos(t + )
2
i-clicker-1 & 2
Youre standing at the end of a springboard, bouncing gently up and down without
leaving the boards surface (performing simple harmonic motion). If you bounce
harder (larger amplitude), the time it takes for each bounce will
A.
become shorter
B.
become the longer
C.
remain same
How about if your friend walks up and bounces with you?
Black board example 15.1
An mass oscillates with an amplitude of 4.00 m, a
frequency of 0.5 Hz and a phase angle of /4.
(a)
(b)
(c)
What is the period T?
Write an equation for the displacement of the
particle.
Calculate the velocity and acceleration of the
object at any time t.
(d) Determine the position, velocity and acceleration of the object at time t = 1.00s.
(e) Calculate the maximum velocity and acceleration of the object.
The block-spring system
2
m
T=
= 2
k
1
1
f= =
T 2
k
m
The frequency depends only on:
- the mass of the block
- the force constant of the spring.
The frequency does not depend on the amplitude.
Black board example 15.2
i-clicker 3
A spring stretches by 3.90 cm when a 10.0 g
mass is hung from it. A 25.0 g mass
attached to this spring oscillates in simple
harmonic motion.
(a)
Calculate the period of the motion.
(b)
Calculate frequency and the angular velocity of the motion.
A) 1.00 s B) 1.21 s C) 1.43 s D) 1.50 s E) 1.59 s
Energy of harmonic oscillator
Kinetic energy:
121
K = mv = m 2 A2 sin 2 (t + )
2
2
Potential energy:
1212
U = kx = kA cos 2 (t + )
2
2
Total energy:
E = K +U =
121
kA = mvmax 2 = constant
2
2
Black board example 15.3
A 0.200 kg mass is attached to a spring
and undergoes simple harmonic
motion with a period of 0.250 s. The
total energy of the system is 2.00 J.
(a)
What is the force constant of the spring?
(b)
What is the amplitude of the motion?
(c)
What is the velocity of the mass when the displacement is 8.00 cm?
(d)
What is the kinetic and potential energy of the system when the displacement is 8.00 cm?
The
(mathematical)
pendulum
(point mass)
2
L
T=
= 2
g
For small motion (less than about
10).
The physical
pendulum
2
I
T=
= 2
mgd
For small motion (less than about
10
I moment of interia
m mass of object
g acceleration due to gravity
d distance from pivot point to center of mass
Black board example 15.4
Find the period of a 14.7
inch (0.37 m) long stick
that is pivoted about one
end and is oscillating in a
vertical plane.
Simple harmonic motion and uniform circular motion
x - component : x(t ) = R cos = R cos t
y - component : y (t ) = R sin = R sin t
Damped, simple harmonic motion
x(t ) = [ A cos( ' t + )] e
k
b2
' =
m 4m 2
bt
b is damping
constant
2m
Forced (Driven) Oscillations and Resonance
b is damping
constant
A damped, harmonic oscillator (ang. frequency ) is driven by an outside,
sinusoidal force with ang. frequency d
Resonance when d = (can get very large amplitudes)