Oscillations

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Oscillations

• Unit 7

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Lesson 1 Simple Harmonic Motion
Fs is a restoring force because it always points
toward the equilibrium position (x 0)
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Applying Newtons Second Law
SF max
-kx max
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Example 1
A block on the end of a spring is pulled to
position x A and released. In one full cycle of
its motion, through what total distance does it
travel ?
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dv
d2x
,
a

Since
dt
dt2
If we call k/m w2,
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(second-order differential equation)
We need a function x(t) whose second derivative
is the same as the original function with a
negative sign and multiplied by w2.
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Proof
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x(t) A cos(wt f)
Amplitude (A) maximum value of the position of
the particle in either the positive or negative
direction.
Angular Frequency (w) number of oscillations
per second.
Since k/m w2,
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Phase Constant (f) initial phase angle. This
is determined by the position of the particle at
t 0.
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Pen traces out cosine curve x(t) A cos(wt f)
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Period (T) the time interval required for the
particle to go through one full cycle of its
motion.
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Frequency (f) the inverse of period. The
number of oscillations that the particle
undergoes per second.
Since T 2p/w,
The Hertz (Hz) is the SI unit for frequency.
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Since f w/2p,
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Velocity in Simple Harmonic Motion
Acceleration in Simple Harmonic Motion
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Maximum Speed in Simple Harmonic Motion
Since sine and cosine oscillate between /- 1,
vmax /- wA
Maximum Acc. in Simple Harmonic Motion
amax /- w2A
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position vs. time
when x at max or min, v 0
when x 0, v is max
velocity vs. time
when x at max, a is max in opposite direction
acceleration vs. time
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Example 2
a) Determine the amplitude, frequency, and
period of the motion.
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b) Calculate the velocity and acceleration of
the object at any time t.
c) Using the results of part b, determine the
position, velocity, and acceleration of the
object at t 1.00 s.
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d) Determine the maximum speed and maximum
acceleration of the object.
e) Find the displacement of the object between t
0 and t 1.00 s.
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Example 3
a) Find the period of its motion.
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b) Determine the maximum speed of the block.
c) What is the maximum acceleration of the block
?
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d) Express the position, speed, and acceleration
as functions of time.
e) The block is released from the same initial
position, xi 5.00 cm, but with an initial
velocity of vi -0.100 m/s. Which parts of the
solution change and what are the new answers for
those that do change ?
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Example 4 AP 1989 3
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a) Determine the speed of the block at the
instant it hits the end of the spring.
b) Determine the period of the simple harmonic
motion that ensues.
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c) Determine the distance that the spring is
compressed at the instant the speed of the
block is maximum.
d) Determine the maximum compression of the
spring.
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e) Determine the amplitude of the simple
harmonic motion.
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Example 5 AP 2003 2
An ideal spring is hung from the ceiling and a
pan of mass M is suspended from the end of the
spring, stretching it a distance D as shown
above. A piece of clay, also of mass M, is then
dropped from a height H onto the pan and sticks
to it. Express all algebraic answers in terms of
the given quantities and fundamental constants.
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a) Determine the speed of the clay at the instant
it hits the pan.
b) Determine the speed of the pan just after the
clay strikes it.
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c) Determine the period of the simple harmonic
motion that ensues.
d) Determine the distance the spring is stretched
(from its initial unstretched length) at the
moment the speed of the pan is a maximum.
Justify your answer.
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The clay is now removed from the pan and the pan
is returned to equilibrium at the end of the
spring. A rubber ball, also of mass M, is dropped
from the same height H onto the pan, and after
the collision is caught in midair before hitting
anything else.
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Lesson 2 Energy in Simple Harmonic Motion
Since v -wA sin(wt f),
KE ½ mw2A2 sin2(wt f)
Since x A cos(wt f),
U ½ kA2 cos2(wt f)
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Total Mechanical Energy of Simple Harmonic
Oscillator
E KE U
E ½ mw2A2 sin2(wt f) ½ kA2 cos2(wt f)
Since w2 k/m,
E ½ kA2 sin2(wt f) cos2(wt f)
Since sin2q cos2q 1,
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The total mechanical energy of a simple harmonic
oscillator is a constant of the motion and is
proportional to the square of the amplitude.
U is small when KE is large, and vice versa.
KE U constant
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E ½ kA2
Since E KE U,
½ kA2 ½ mv2 ½ kx2
Solving for v,
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Example 1
The amplitude of a system moving in simple
harmonic motion is doubled. Determine the change
in the
a) total energy
b) maximum speed
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c) maximum acceleration
d) period
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Example 2
A 0.500 kg cart connected to a light spring for
which the force constant is 20.0 N/m oscillates
on a horizontal, frictionless air track.
a) Calculate the total energy of the system and
the maximum speed of the cart if the amplitude
of the motion is 3.00 cm.
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b) What is the velocity of the cart when the
position is 2.00 cm ?
c) Compare the kinetic and potential energies of
the system when the position is 2.00 cm ?
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Lesson 3 Comparing Simple Harmonic Motion with
Uniform Circular Motion
As the turntable rotates with constant angular
speed, the shadow of the ball moves back and
forth in simple harmonic motion.
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Reference Circle
Simple harmonic motion along a straight line can
be represented by the projection of uniform
circular motion along a diameter of a reference
circle.
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Example 1
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Example 2
a) Explain why the bump, from your viewpoint
behind the car, executes simple harmonic
motion.
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b) If the radii of the cars tires are 0.300 m,
what is the bumps period of oscillation ?
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Lesson 4 The Pendulum
Forces acting on bob
Tension in string
Gravitational force mg
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Ft mg sinq always acts opposite to the
displacement of the bob
Since s Lq and L is constant,
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Instead of x A cos(wt f),
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(Period and frequency are independent of mass.)
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Example 1
Christian Huygens (1629-1695), the greatest
clockmaker in history, suggested that an
international unit of length could be defined as
the length of a simple pendulum having a period
of exactly 1s.
a) How much shorter would our length unit be had
his suggestion been followed ?
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b) What if Huygens had been born on another
planet ? What would the value of g have to be on
that planet such that the meter based on
Huygens pendulum would have the same value as
our meter ?
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Example 2
A simple pendulum has a mass of 0.250 kg and a
length of 1.00 m. It is displaced through an
angle of 15.0o and released. What is the
a) maximum speed
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b) maximum angular acceleration
c) maximum restoring force ?
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Example 3
A simple pendulum is 5.00 m long.
a) What is the period of small oscillations for
this pendulum if it is located in an elevator
accelerating upward at 5.00 m/s2 ?
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b) What is its period if the elevator is
accelerating downward at 5.00 m/s2 ?
c) What is the period of this pendulum if it is
placed in a truck that is accelerating
horizontally at 5.00 m/s2 ?
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The Physical Pendulum
A hanging object that oscillates about a fixed
axis that does not pass through its center of
mass and the object cannot be approximated as a
point mass.
Gravitational force produces a torque about an
axis through O.
t mgd sinq
Since St Ia,
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If q is small so that sinq is almost q,
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Since the period of a pendulum is
2p
T
,
w
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Example 4
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Torsional Pendulum
t -kq
k is called the torsion constant of the wire
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d2q
,
Since St I
dt2
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Example 5
A torsional pendulum is formed by taking a meter
stick of mass 2.00 kg, and attaching to its
center a wire. With its upper end clamped, the
vertical wire supports the stick as the stick
turns in a horizontal plane. If the resulting
period is 3.00 minutes, what is the torsion
constant for the wire ?