Transcript Simple Harmonic Motion

Simple Harmonic Motion
ISAT 241
Fall 2004
David J. Lawrence
Simple Harmonic Motion

Mass Attached to a Spring
x<0
m
x = 0 “Equilibrium Position”
x>0
x
Simple Harmonic Motion

Mass Attached to a Spring
– When the mass is stationary, it is in its
equilibrium position (Net force = 0).
– If we pull the mass down slightly and then release
it, what happens?
– Graph the displacement from equilibrium as a
function of time.
• x(t) = A cos (w t)
• where A = amplitude = a constant, and
• w = angular frequency = a constant.
• We’ll learn more about these quantities soon.
Serway & Jewett, Principles of Physics
Figure 12.1
Brooks/Cole – Thomson
Learning
Simple Harmonic Motion
A
Displacement
x(t)
(m)
0
Time
(s)
-A
wA
Velocity
v(t)
(m/s)
0
Time
(s)
-wA
w2A
Acceleration
a(t)
(m/s2)
0
-w2A
Time
(s)
Simple Harmonic Motion

Equations for this Special Case
– Displacement from Equilibrium:
x(t) = A cos (w t)
– Velocity:
dx
v(t) =
= - w A sin (w t)
dt
dv
– Acceleration: a(t) =
=
dt
- w2 A cos (w t)
Simple Harmonic Motion
Definitions of Terms
• Amplitude = A = the maximum displacement of the
moving object from its equilibrium position.
• (unit = m)
• Period = T = the time it takes the object to complete
one full cycle of motion.
• (unit = s)
• Frequency = f = the number of cycles or vibrations
per unit of time.
• (unit = cycles/s = 1/s = Hz = hertz)
Label This Graph !
0
Time
(s)
Simple Harmonic Motion

Definitions of Terms (continued)
– Angular Frequency = w
(unit = radians/s = rad/s)

2π
ω  2π f 
T
– Phase Constant = Phase Angle = f
 (unit = radians)
 In general, simple harmonic motion cannot be described
by a “pure” sine or cosine function, so a phase constant,
f , or phase angle must be introduced.
x(t) = A cos (w t + f)

E.g.,

(wt + f) is called the phase of the motion
Summary Graphs for SHM –
General Case
In the most
general case, the
displacement
graph doesn’t
begin at a peak.
This means that
f  
Simple Harmonic Motion

General Equations
– Displacement from Equilibrium:
x(t) = A cos (w t + f)
– Velocity:
– Acceleration:
v(t) =
a(t) =
dx
dt
dv
dt
= - wA sin (w t + f)
2A cos (w t + f)
w
=
Simple Harmonic Motion -- Example

An object oscillates with SHM along the x-axis. Its
displacement varies with time according to the equation

x  x ( t )  (4.0m) cos( t + )
4
where t is in seconds and the two angles in parentheses are
in radians. (See figure on next slide.)
(a) Determine the amplitude, phase constant, angular
frequency, frequency, and period of the motion.
(b) Calculate the velocity and acceleration of the object at any
time t.
(c) Determine the position, velocity, and acceleration of the
object at t = 1 s.
Serway & Jewett, Principles of Physics
Figure 12.1
Brooks/Cole – Thomson
Learning
S. H. M. – Example (continued)

An object oscillates with SHM along the x-axis. Its
displacement varies with time according to the equation
x  x( t )  ( 4.0m) cos( t +

4
)
where t is in seconds and the two terms in parentheses are in
radians.
(d) Determine the maximum displacement from the origin,
maximum speed, and maximum acceleration of the object.
(e) Find the displacement of the object between t = 0 and
= 1 s.
(f ) What is the phase of the motion at t = 2.00 s?
t
Simple Harmonic Motion

Important properties of an object moving in
simple harmonic motion:
– The displacement, velocity, and acceleration all
vary sinusoidally with time, but are not in phase.
– The acceleration is proportional to the
displacement, but in the opposite direction.
– The frequency and period of the motion are
independent of the amplitude.
Simple Harmonic Motion

Whenever the force acting on an object is linearly
proportional to the displacement and in the opposite
direction, the object exhibits simple harmonic
motion.

We have been considering the simple example of a
mass attached to a spring.
Mass Attached to a Spring
 Hooke’s Law:
Fspring = - kx
 Newton’s Second Law: SF = ma = m (d2x/dt2)
 Therefore,
(d2x/dt2) = - (k/m) x
 This is a differential equation, which can be
solved for x(t).
 The solution is the equation we have been
using all along:

x(t) = A cos (w t + f)
 Show that this is a solution to the differential
equation by substitution.
Mass Attached to a Spring

For this case, the angular frequency is
w

k
m
so the frequency and period are
1 w
1 k
f
 
T 2 2 m
m
1 2
 2
T 
w
k
f
Mass-Spring System -- Example 1

Car hitting a pothole in the road.
Mass-Spring System -- Example 2

A 200 g mass is connected to a light spring with
force constant 5 N/m, and is free to oscillate on a
horizontal, frictionless surface. The mass is
displaced 5 cm to the right from equilibrium and
released from rest. (See figure on next slide.)
(a) Find the period of the mass’ motion.
(b) Find the displacement, speed, and
acceleration as functions of time.
(c) Determine the max. speed of the mass.
(d) Determine the max. acceleration of the mass.
Summary Graphs for SHM –
General Case
In the most
general case, the
displacement
graph doesn’t
begin at a peak.
This means that
f  