Oscillatory Motion

1
Oscillatory Motion

• Object attached to a spring
• Simple harmonic motion
• Energy of a simple harmonic oscillator
• Simple harmonic motion and circular motion
• The pendulum

2
An Object Attached to a Spring
When acceleration is proportional to and in the
opposite direction of the displacement from
equilibrium, the object moves with Simple
Harmonic Motion.
3
Equation of Motion
Second order differential equation for the motion
of the block
The harmonic solution for the spring-block system
where
4
Some Terminology
Angular frequency
Phase constant

Phase
Amplitude
5
Properties of Periodic Functions
f / w

• The function is periodic with T.
• The maximum value is the amplitude.

6
Simple Harmonic Motion
7
Properties of Simple Harmonic Motion

• Displacement, velocity and acceleration are
  sinusoidal with the same frequency.
• The frequency and period of motion are
  independent of the amplitude.
• Velocity is 90 out-of-phase with displacement.
• Acceleration is proportional to displacement but
  in the opposite direction.

8
Example P15.10

• A piston in a gasoline engine is in simple
  harmonic motion. If the extremes of its position
  relative to its center point are 5.75 cm, find
  the maximum velocity and acceleration of the
  piston when the engine is running at the rate of
  3750 rev/min.

9
The Block-Spring System
Frequency is only dependent on the mass of the
object and the force constant of the spring
10
Example 15.3
11
Energy of the Harmonic Oscillator

• Consider the block-spring system.
• If there is no friction, total mechanical energy
  is conserved.
• At any given time, this energy is the sum of the
  kinetic energy of the block and the elastic
  potential energy of the spring.
• Their relative share of the total energy
  changes as the block moves back and forth.

12
Energy of the Harmonic Oscillator
13
Energy of the Harmonic Oscillator
t x v a K U
0 A 0 -w2A 0 ½kA2
T/4 0 -wA 0 ½kA2 0
T/2 -A 0 w2A 0 ½kA2
3T/4 0 wA 0 ½kA2 0
T A 0 -w2A 0 ½kA2
14
Example P15.18

• A block-spring system oscillates with an
  amplitude of 3.70 cm. The spring constant is 250
  N/m and the mass of the block is 0.700 kg.
• Determine the mechanical energy of the system.
• Determine the frequency of oscillation.
• If the system starts oscillating at a point of
  maximum potential energy, when will it have
  maximum kinetic energy?
• When is the next time it will have maximum
  potential energy?

15
The Simple Pendulum
The tangential component of the gravitational
force is a restoring force
For small q (q lt 10)
The form as simple harmonic motion
16
The Physical Pendulum
17
Example 15.7
18
Simple Harmonic Motion and Uniform Circular Motion
19
Damped Oscillations

• Suppose a non-conservative force (friction,
  retarding force) acts upon the harmonic
  oscillator.

20
Review

• Restoring forces can result in oscillatory
  motion.
• Displacement, velocity and acceleration all
  oscillate with the same frequency.
• Energy of a harmonic oscillator will remain
  constant.
• Simple harmonic motion is a projection of
  circular motion.
• Resistive forces will dampen the oscillations.