Transcript Document

Simple Harmonic Motion
In periodic motion, a body repeats a certain motion
indefinitely, always returning to its starting point after a
constant time interval and then starting a new cycle.
Every system that has this force exhibits a simple harmonic
motion (SHM) and is called a simple harmonic oscillator.
Simple harmonic motion is periodic motion in the
absence of friction and produced by a restoring force
that is directly proportional to the displacement and
oppositely directed.
x
m
F
A restoring force, F, acts in
the direction opposite the
displacement of the
oscillating body.
F = -kx
Hooke’s Law
A mass m attached to a spring executes SHM when the
spring is pulled out and released. The spring’s PE
becomes K as the mass begins to move, and the K of the
mass becomes PE again as its momentum causes the
spring to overshoot the equilibrium position and become
compressed.
Displacement in SHM
x
m
x = -A
x=0
x = +A
• The maximum displacement is called
the amplitude A.
Velocity in SHM
v (-)
v (+)
m
x = -A
x=0
x = +A
• Velocity is zero at the end points and a
maximum at the midpoint in either direction
Acceleration in SHM
+a
-x
+x
-a
m
x = -A
x=0
x = +A
F  ma   kx
• Acceleration is a maximum at the end points
and it is zero at the center of oscillation.
Acceleration vs. Displacement
a
v
x
m
x = -A
x=0
x = +A
Given the spring constant, the displacement, and
the mass, the acceleration can be found from:
F  ma   kx or
 kx
a
m
Note: Acceleration is always opposite to displacement.
Conservation of Energy
The total mechanical energy (PEs + K) of a SH
system is constant.
a
v
x
A
B
m
x = -A
x=0
x = +A
For any two points A and B, we may write:
½mvA2 + ½kxA 2 = ½mvB2 + ½kxB 2
GRAPH OF SHM
The graph shown below depicts the up and down
oscillation of the mass at the end of a spring.
One complete cycle is from a to b, or from c to d.
The time taken for one cycle is T, the period.
PERIOD AND FREQUENCY
The period T of a body of mass m attached to
a spring of force constant k
UNITS:
T in seconds
f in Hz (s-1)
Simple Pendulum
For small displacements a pendulum obeys SHM.
Its period is:
The period and frequency DO NOT depend on the mass.
6.1 For the motion shown in the figure, find:
a. Amplitude
b. Period
c. Frequency
a. Amplitude: maximum displacement from equilibrium
A = +- 0.75 cm
b. T = time for one complete cycle
T = 0.2 s
c. f = 1/T = 1/0.2 = 5 Hz
6.2 A 200-g mass vibrates horizontally without friction at the end
of a horizontal spring for which k = 7.0 N/m. The mass is
displaced 5.0 cm from equilibrium and released. Find:
COE
a. Maximum speed
m = 0.2 kg
k = 7 N/m
xo = 0.05 m
1
1
1
kx  mv  kx
2
vmax is at x = 0 then:
2
o
2
v  x0
2
2
2
k
7
 0.05
0.2
m
v = 0.295 m/s
b. Speed when it is 3.0 cm from equilibrium.
x = 0.03 m
1
2
1
1
kx  mv  kx
2
o
2
2
2
2
k
7
2
2
2
2
v
( xo  x ) 
0.05  0.03 

0.2
m
v = 0.236 m/s
c. What is the acceleration in each of these cases?
F = ma = - kx
 kx
a
m
a. x = 0 therefore a = 0
b. x = 0.03 m therefore
7(0.03)
a
= - 1.05 m/s2
0.2
6.3 As shown in the figure, a long, light piece of spring steel is
clamped at its lower end and a 2.0-kg ball is fastened to its top
end. A horizontal force of 8.0 N is required to displace the ball 20
cm to one side as shown. Assume the system to undergo SHM
when released. Find:
SHM
a. The force constant of the spring
F=8N
x = 0.2 m
m = 2 kg
F
8
k

= 40 N/m
0.2
x
b. Find the period with which the ball will vibrate back and forth.
2
m
 2
T  2
= 1.4 s
40
k
6.3 In a laboratory experiment a student is given a stopwatch, a
wooden bob, and a piece of cord. He is then asked to determine
the acceleration of gravity. If he constructs a simple pendulum of
length 1 m and measures the period to be 2 s, what value will he
obtain for g?
SHM
L=1m
T=2s
L
T  2
g
4 2 L 4 2 (1)
2
g

=
9.86
m/s
2
T2
2
DAMPED HARMONIC MOTION
A system undergoing SHM will exhibit damping.
Damping is the loss of mechanical energy as the
amplitude of motion gradually decreases.
In the mechanical systems studied in the previous
sections, the losses are generally due to air resistance
and internal friction and the energy is transformed into
heat.
For the amplitude of the motion to remain constant, it is
necessary to add enough energy each second to offset
the energy losses due to damping.
In many instances damping is a desired effect. For
example, shock absorbers in a car remove unwanted
vibration.
FORCED VIBRATIONS: RESONANCE
An object subjected to an external oscillatory force
tends to vibrate. The vibrations that result are called
forced vibrations. These vibrations have the same
frequency as the external force and not the natural
frequency of the object.
If the external forced vibrations have the same
frequency as the natural frequency of the object, the
amplitude of vibration increases and the object exhibits
resonance. The natural frequency (or frequencies) at
which resonance occurs is called the resonant
frequency.
EXAMPLES OF RESONANCE