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Lecture 17
Oscillations
Today’s Topics:
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•
•
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Periodic motion (Simple Harmonic Motion)
Springs and pendulums
Energy
Damped and driven motion
Restoring Forces
No restoring force
Restoring force
returns the ball
to equilibrium
ACT: to the center of Earth
A hole is drilled through the
a) you fall to the center and stop
center of Earth and emerges
b) you go all the way through and
continue off into space
on the other side. You jump
into the hole. What happens
to you ?
c) you fall to the other side of
Earth and then return
d) you won’t fall at all
You fall through the hole. When you reach the
center, you keep going because of your inertia.
When you reach the other side, gravity pulls
you back toward the center. Gravity is a
restoring force here!
Follow-up: Where is your acceleration zero?
How do we describe these oscillations
about equilibrium?
x = A cos q
But q = wt , so
x = A cos wt
Simple Harmonic Motion
x = A cos wt
2p
w = 2p f =
T
1
f =
T
amplitude A: the maximum displacement
period T: the time required to complete one cycle
frequency f: the number of cycles per second (measured in Hz)
ACT: simple harmonic motion
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 Δx = 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.
Example
The position of a simple harmonic oscillator is given by x(t ) = (0.50 m) cos (p3 t )
where t is in seconds. What is the period of the oscillator?
x = A cos wt
A = 0.5 m
w=
p
3
rad/s
Velocity and Acceleration
Where is vmax?
ax = - !
Aw cos wt
2
amax
Where is amax?
Springs
HOOKE’S LAW: RESTORING
FORCE OF AN IDEAL SPRING
The restoring force on an ideal
spring is
Fx = - k x
How do we determine ω for a spring?
The frequency is determined by the physical properties of the system
x = A cos w t
a x = - Aw 2 cos w t
å F = -kx = ma
- kA = -mAw
k
w=
m
2
x
ACT: 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.
To measure the mass of an astronaut on the space station they employ a device
that consists of a spring-mounted chair in which the astronaut sits. The spring has
a spring constant of 606 N/m and the mass of the chair is 12.0 kg. The measured
period is 2.41 s. Find the mass of the
astronaut.
k
w=
mtotal
mtotal = k w 2
mchair + mastro
k
=
(2p T )2
mastro
k
=
- mchair
2
(2p T )
(
606 N m )(2.41 s )
=
2
4p
2
- 12.0 kg = 77.2 kg
Springs and Energy
DEFINITION OF ELASTIC POTENTIAL ENERGY
The elastic potential energy is the energy that a spring
has by virtue of being stretched or compressed. For an
ideal spring, the elastic potential energy is
PE elastic = 12 kDx 2
SI Unit of Elastic Potential Energy: joule (J)
Total
Mechanical
Energy
1 2 1
E = mv + kDx 2
2
2
As a function of time,
The total energy is constant; as the kinetic energy
increases, the potential energy decreases, and vice versa.
Since we know the position and velocity as functions of time,
we can find the maximum kinetic and potential energies:
The Pendulum
The restoring force of a
pendulum is proportional to sin θ,
whereas the restoring force for a
spring is proportional to the
displacement (which is θ in this
case).
Frestoring = −mgsin θ = ma
However, for small angles, sinθ and θ are
approximately equal (small angle approximation)
Substituting θ for sin θ allows us to treat the
pendulum in a mathematically identical way to the
mass on a spring. We find that the period of a
pendulum depends on the length of the string and g:
ACT: period of a pendulum
Two pendulums have the
same length, but different
masses attached to the
string. How do their
periods compare?
a) period is greater for the greater mass
b) period is the same for both cases
c) period is greater for the smaller mass
The period of a pendulum depends on the length and the
acceleration due to gravity, but it does not depend on the
mass of the bob.
L
=
2π
T
g
Follow-up: What happens if the amplitude is doubled?
ACT: pendulum on the moon
A swinging pendulum has period
T on Earth. If the same pendulum
a) period increases
were moved to the Moon, how
b) period does not change
does the new period compare to
c) period decreases
the old period?
The acceleration due to gravity is smaller on the Moon. The
relationship between the period and g is given by:
T = 2π
therefore, if g gets smaller, T will increase.
L
g
ACT: grandfather clock
A grandfather clock has a
weight at the bottom of the
pendulum that can be moved
up or down. If the clock is
running slow, what should
you do to adjust the time
properly?
a) move the weight up
b) move the weight down
c) moving the weight will not matter
d) call the repairman
The period of the grandfather clock is too long, so we need to
decrease the period (increase the frequency). To do this, the length
must be decreased, so the adjustable weight should be moved up in
order to shorten the pendulum length.
L
=
T 2π
g
Damped Harmonic Motion
1) simple harmonic motion
2&3) underdamped
4) critically damped
5) overdamped
Driven Harmonic Motion and
Resonance
When a force is applied to an oscillating system at all times,
the result is driven harmonic motion.
Here, the driving force has the same frequency as the
spring system and always points in the direction of the
object’s velocity.
Resonance occurs when the frequency of the force matches a
natural frequency at which the object will oscillate.