Warmup

1
Warmup

• A uniform beam 2.20m long with mass m25.0kg, is
  mounted by a hinge on a wall as shown. The beam
  is held horizontally by a wire that makes a 30
  angle as shown. The beam supports a mass M
  280kg suspended from its end. Determine the
  components of the force FH that the hinge exerts
  and the components tension, FT in the supporting
  wire.

30
M
2
Harmonic Motion
3
Simple Harmonic Motion

• When a vibration or oscillation repeats itself,
  back and forth, over the same path, the motion is
  periodic
• Any vibrating system in which the restoring force
  is directly proportional to the negative of the
  displacement is said to exhibit simple harmonic
  motion
• Such a system is often called a simple harmonic
  oscillator

A mass oscillating on the end of a spring is an
example of a simple harmonic oscillator
4
The Motion of an Oscillating Object

• The motion of an object undergoing simple
  harmonic oscillation is sinusoidal in nature

x(t) A sin (2pt/T)
5
The Total Energy of a Vibrating System is Constant

• KE PE constant
• If the maximum amplitude of the motion is x0 then
  the energy at any point x is given by
• ½ mv2 ½ kx2 ½ kx02
• From this we can solve for velocity
• v v (x02 x2)(k/m)
• From Hookes law, F -kx and F ma, therefore
• a -(k/m) x

6
Reference Circle

• Point P moves with constant velocity v0 around
  the circle
• Point A is the projection of point P on the x
  axis
• The motion of A back and forth about point O is
  second harmonic motion
• The time for P to go around the circle is T
• The velocity of point A is
• v -v0 sin ?

Once around in time T
v0
?
P
v
r x0
?
O
A
Displacement x
7
Reference Circle

• The period T is
• T 2pr0/v0 2px0/v0
• But v0 is the maximum speed of point A
• Maximum speed occurs when x 0
• Therefore, v0 x0 v(k/m)
• Which makes T 2pv(m/k)
• And, since f 1/T,
• f (1/2p)v(k/m)

8
Period and Frequency of an Oscillator

• The period of a simple harmonic oscillator is
  given by
• T 2pv(m/k)
• Therefore, since f 1/T,
• f (1/2p)v(k/m)

9
Simple Pendulums

• For a pendulum the restoring force is
• F -mg sin ?,
• But for small displacements,
• F -mg sin ? -mg?
• But since x L? we have
• F (mg/L) x
• Therefore the motion is essentially harmonic

10
Simple Pendulums

• But if a pendulum is an harmonic oscillator, then
  k mg/L, therefore
• T 2pv (m/mg/L) 2pv(L/g)
• and
• F 1/T 1/(2p)v(g/L)
• The frequency of a pendulum does not depend on
  the mass of the pendulum bob!
• Consider the case of large and small children on
  a swingthe period remains the same

11
In Real Life Most Harmonic Motion is Damped

• Natural oscillations decrease with time due to
  friction and other losses
• Sometimes the damping is so great that the motion
  does not even appear to be harmonic

A overdamped B critically damped C
underdamped
A shock absorber is a damped oscillator
12
Shock Absorbers Keep a Car From Oscillating
13
Links

• Java simulationspring
• Java simulation-oscillator
• Simple Harmonic Motion and Uniform Circular
  Motion

14
Do Now (10/8/13)

• Suppose that a pendulum has a period of 1.5
  seconds. How long does it take to make a complete
  back-and-forth vibration? Is this 1.5 s period
  pendulum longer or shorter in length than a 1 s
  period pendulum?

15
Practice

• Brainstorm answers to the Conceptual Questions in
  Chapter 13 (10 min)
• Complete the Multiple Choice questions in Chapter
  13 before the end of class.

16
Do Now (10/9/13)

• Pass in your HW then wait quietly for instructions