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Lecture 18 – Oscillations
about Equilibrium
Periodic Motion
Period: time required for one cycle of periodic motion
Frequency: number of oscillations per unit time
This unit is called
the Hertz:
Simple Harmonic Motion
A spring exerts a restoring force that is
proportional to the displacement from
equilibrium:
Displaced,
at rest
Moving, past
equilibrium point
Displaced,
at rest
Moving, past
equilibrium point
Displaced,
at rest
This is called “Simple
Harmonic Motion”
Simple Harmonic Motion
A mass on a spring has a displacement as a
function of time that is a sine or cosine curve:
Here, A is called
the amplitude of
the motion.
Simple Harmonic Motion
If we call the period of the motion T – this is the
time to complete one full cycle – we can write the
position as a function of time:
Time
Position
t =0
x=A
t=T
x = Acos(2π) = A
t = T/2 x = Acos(π) = -A
T
t = T/4 x = Acos(π/2) = 0
The position at time t +T is the same as the
position at time t, as we would expect.
Sine vs Cosine
x at t=0 : A
v at t=0 : 0
x at t=0 : 0
v at t=0 : >0
Harmonic Motion I
A mass on a spring in SHM has
amplitude A and period T. What
is the total distance traveled by
the mass after a time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
Harmonic Motion I
A mass on a spring in SHM has
amplitude A and period T. What
is the total distance traveled by
the mass after a time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
In the time interval T (the period), the mass goes through
one complete oscillation back to the starting point. The
distance it covers is A + A + A + A = (4A).
Harmonic Motion II
A mass on a spring in SHM has
amplitude A and period T. What is the
net displacement of the mass after a
time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
Harmonic Motion II
A mass on a spring in SHM has
amplitude A and period T. What is the
net displacement of the mass after a
time interval T?
a) 0
b) A/2
c) A
d) 2A
e) 4A
The displacement is x = x2 – x1. Because the
initial and final positions of the mass are the
same (it ends up back at its original position),
then the displacement is zero.
Follow-up: What is the net displacement after a half of a period?
The Pendulum
A simple pendulum consists of a mass m (of
negligible size) suspended by a string or rod of
length L (and negligible mass).
The angle it makes with the vertical varies with
time as a sine or cosine.
How a pendulum is like the mass on a spring
Looking at the forces
on the pendulum bob,
we see that the
restoring force is
proportional to sin θ
The restoring force for
a spring is proportional
to the displacement
This is the condition for
simple harmonic motion
Approximation for sin θ
However, for small angles, sin θ and θ are
approximately equal.
θ (deg)
θ (rad)
sin(θ)
1
0.01745 0.01745
5
0.08727 0.08716
10
0.1745
0.1736
20
0.3491
0.3420
Pendulum for small angles = simple harmonic
for small angles of the
pendulum bob, the
restoring force is
proportional to θ
F = -mg θ = -mg s / L
The restoring force for a
spring is proportional to
the displacement
F= -kx
So: the motion of the angle of the pendulum is the same as
the motion for the mass on a spring, with k
mg/L
Uniform Circular Motion and Simple Harmonic
Motion
An object in simple
harmonic motion has the
same motion as one
component of an object in
uniform circular motion
Assume oscillation of mass on the spring has
the same period T as the circular motion of the
peg on the “record player”
Position of Peg in Circular Motion
Here, the object in circular motion has
an angular speed of:
where T is the period of motion
of the object in simple harmonic
motion.
The position as a function of time:
... just like the simple harmonic motion!
Velocity of Peg in Circular Motion
Linear speed v = Aω
x component:
Acceleration of Peg in Circular Motion
Linear acceleration a = Aω2
x component:
Summary of Simple Harmonic Motion
The position as a function of time:
From this comparison with circular motion, we can see:
The angular frequency:
The velocity as a function of time:
The acceleration as a function of time:
Speed and Acceleration
A mass on a spring in SHM has
a) x = A
amplitude A and period T. At
b) x > 0 but x < A
what point in the motion is v = 0
c) x = 0
and a = 0 simultaneously?
d) x < 0
e) none of the above
Speed and Acceleration
A mass on a spring in SHM has
a) x = A
amplitude A and period T. At
b) x > 0 but x < A
what point in the motion is v = 0
c) x = 0
and a = 0 simultaneously?
d) x < 0
e) none of the above
If both v and a were zero at
the same time, the mass
would be at rest and stay at
rest! Thus, there is NO
point at which both v and a
are both zero at the same
time.
Follow-up: Where is acceleration a maximum?
The Period of a Mass on a Spring
For the mass on a spring:
Substituting the time dependencies of a and x gives
and the period is:
The Period of a Mass on a Spring
Vertical Spring
Fs=kx
What if the mass hangs from a vertical spring?
new equilibrium position: x= -d = -mg/k
total force as a function of x:
d
x=0
x=-d
W=mg
with
Looks like the same spring, with a different
equilibrium position (x’=0 -> x = -d)
Simple harmonic motion is unchanged from the horizontal case!
Energy Conservation in Oscillatory Motion
In an ideal system with no nonconservative forces,
the total mechanical energy is conserved. For a
mass on a spring:
Since we know the position and velocity as
functions of time, we can find the maximum
kinetic and potential energies:
Energy Conservation in Oscillatory Motion
As a function of time,
So the total energy is constant; as the kinetic
energy increases, the potential energy decreases,
and vice versa.
0
Period of a Pendulum
A pendulum is like the mass on a spring, with k=mg/L
Therefore, we find that the period of a pendulum
depends only on the length of the string:
Physical Pendula
A physical pendulum is a
solid mass that oscillates
around its center of mass, but
cannot be modeled as a point
mass suspended by a
massless string. Examples:
Period of a Physical Pendulum
In this case, it can be shown that the period
depends on the moment of inertia:
Substituting the moment of inertia of a point mass a
distance l from the axis of rotation gives, as expected:
Damped Oscillations
In most physical situations, there is a
nonconservative force of some sort, which will
tend to decrease the amplitude of the oscillation,
and which is typically proportional to the speed:
This causes the amplitude to decrease
exponentially with time:
Damped Oscillations
This exponential decrease is shown in the
figure:
“underdamped” means that there is
more than one oscillation
Damped Oscillations
The previous image shows a system that is
underdamped – it goes through multiple
oscillations before coming to rest.
A critically damped system is one that relaxes
back to the equilibrium position without
oscillating and in minimum time;
an overdamped system will also not oscillate
but is damped so heavily that it takes longer to
reach equilibrium.
Driven Oscillations and Resonance
An oscillation can be driven by an oscillating driving
force; the frequency of the driving force may or may
not be the same as the natural frequency of the
system.
Driven Oscillations and Resonance
If the driving frequency
is close to the natural
frequency, the amplitude
can become quite large,
especially if the damping
is small. This is called
resonance.
Energy in SHM I
A mass on a spring oscillates in
simple harmonic motion with
amplitude A. If the mass is
doubled, but the amplitude is not
changed, what will happen to the
total energy of the system?
a) total energy will increase
b) total energy will not change
c) total energy will decrease
Energy in SHM I
A mass on a spring oscillates in
simple harmonic motion with
amplitude A. If the mass is
doubled, but the amplitude is not
changed, what will happen to the
total energy of the system?
a) total energy will increase
b) total energy will not change
c) total energy will decrease
The total energy is equal to the initial value of the
elastic potential energy, which is PEs = kA2. This
does not depend on mass, so a change in mass will
not affect the energy of the system.
Follow-up: What happens if you double the amplitude?
Spring on the Moon
A mass oscillates on a vertical
spring with period T. If the whole
setup is taken to the Moon, how
does the period change?
a) period will increase
b) period will not change
c) period will decrease
Spring on the Moon
A mass oscillates on a vertical
spring with period T. If the whole
setup is taken to the Moon, how
does the period change?
a) period will increase
b) period will not change
c) period will decrease
The period of simple harmonic motion depends only on the
mass and the spring constant and does not depend on the
acceleration due to gravity. By going to the Moon, the value of
g has been reduced, but that does not affect the period of the
oscillating mass–spring system.