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Oscillatory Motion

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.

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Oscillatory Motion

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  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. Angular Frequency (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

  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

  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 < 10°): The form as simple harmonic motion

  16. For small q (q < 10°): 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.

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