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Oscillatory Motion. Object attached to a spring Simple harmonic motion Energy of a simple harmonic oscillator Simple harmonic motion and circular motion The pendulum. An Object Attached to a Spring.

Oscillatory Motion

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Oscillatory Motion

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

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.

Equation of Motion

Second order differential equation for the motion of the block

The harmonic solution for the spring-block system

where

(rad/s)

Period

(s)

Frequency

(1/s=Hz)

Properties of Periodic Functions- The function is periodic with T.
- The maximum value is the amplitude.

f / w

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.

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.

The Block-Spring System

Frequency is only dependent on the mass of the object and the force constant of the spring

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.

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?

The Simple Pendulum

The tangential component of the gravitational force is a restoring force

For small q (q < 10°):

The form as simple harmonic motion

For small q (q < 10°):

The Physical PendulumDamped Oscillations

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

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.