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Whenever the force acting on an object is: Proportional to the displacement

Whenever the force acting on an object is: Proportional to the displacement In the opposite direction, the object exhibits simple harmonic motion (SHM) . Examples mass attached to a spring simple pendulum. Definitions of Terms

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Whenever the force acting on an object is: Proportional to the displacement

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  1. Whenever the force acting on an object is: • Proportional to the displacement • In the opposite direction, the object exhibits simple harmonic motion (SHM). Examples • mass attached to a spring • simple pendulum.

  2. Definitions of Terms • Amplitude = A = the maximum displacement of the moving object from its equilibrium position. • Period = T = the time it takes the object to complete one full cycle of motion. • Frequency = f = the number of cycles or vibrations per unit of time.

  3. Mass Attached to a Spring x > 0 m x = 0 “Equilibrium Position” x < 0 Vertical Spring x = displacement from equilibrium

  4. Period of an object on a vertical spring exhibiting SHM is: T = period m = mass of object K = spring constant

  5. Force always opposite the displacement from equilibrium Horizontal Spring • If we stretch a spring with a mass on the end and let it go, the mass will oscillate back and forth (if there is no friction). • This oscillation is called Simple HarmonicMotion, because F is a restoring force.

  6. X F As previously stated, a simple harmonic oscillator is any object that oscillates and is subject to a restoring force. Example: horizontal mass on the end of a spring. F is a linear restoring force. Hooke’s law F= -kx applies

  7. The frequency and period of the simple harmonic oscillator are independent of the amplitude.

  8. Sine or cosine curve representation of a restoring force and simple harmonic motion. t

  9. Another View

  10. Case 2 - The Simple Pendulum. A component of the weight acts as the restoring force Component of weight restoring mass to equilibrium mg sin

  11. Period for The Simple Pendulum: A pendulum is made by suspending a mass m at the end of a string of length L. The period of oscillation for small displacements is given by the following formula. T = period L = length “g”= acceleration due to gravity

  12. Period of a simple harmonic oscillator representation in the form of a cosine curve. T/4 = time for quarter cycle T/2 = time for half cycle 3T/2 = time for three quarters of a cycle

  13. Simple Harmonic Motion and Circular Motion, compared to circular motion

  14. Energy in Simple Harmonic Motion

  15. Mass-Spring System -- Example 1 Car hitting a pothole in the road.

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