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Simple Harmonic Motion: SHM

Simple Harmonic Motion: SHM. Linear Restoring Force:. Ideal Spring. Spring Constant k. Energy Conservation:. Amplitude A. constant (independent of time). At x max = A v x = 0 hence. (true at any time t). The maximum speed v max occurs when x = 0. General Solution!.

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Simple Harmonic Motion: SHM

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  1. Simple Harmonic Motion: SHM • Linear Restoring Force: Ideal Spring Spring Constant k • Energy Conservation: Amplitude A constant (independent of time) At xmax = A vx = 0 hence (true at any time t) The maximum speed vmax occurs when x = 0. General Solution! The phase angle f determines where the mass m is at t = 0, x(t=0) = Acosf. If x(t=0) = A then f = 0. PHY 2053

  2. Uniform Circular Motion & SHM • Uniform Circular Motion: Project uniform circular motion (constant angular velocity w) of a vector with length A onto the x-axis and you get SHM! • Simple Harmonic Motion (SHM): Amplitude x(t) If x(t=0) = A then The period T is the time is takes for one circular revolution: Time t w = angular frequency (in rad/sec) f = frequency (in Hz) T = period (in s) PHY 2053

  3. SHM: Graphical Representation If x(t=0) = A then PHY 2053

  4. SHM: General Solution If the acceleration ax(t) and the position x(t) are related as follows: where C is some constant then If x(t=0) = A then f = 0: If x(t=0) = 0 and vx(t=0) > 0 then f = p/2: PHY 2053

  5. SHM: General Solution • Angular Oscillations SHM: If the angular acceleration a(t) and the angular position q(t) are related as follows: where C is some constant then PHY 2053

  6. The Pendulum: Small Oscillations SHM • Simple Pendulum: Small pendulum bob with mass m on string of lengh L and negligible mass. Calculate the torque about the axis of rotation as follows: SHM with period T given by (simple pendulum) • Physical Pendulum: Moment of inertia, I, Length L, mass m, distance from axis of rotation to the center-of-mass, dcm. Calculate the torque about the axis of rotation as follows: SHM with period T given by (physical pendulum) PHY 2053

  7. SHM: Example Problems • A simple harmonic oscillator consists of a block of mass 2 kg attached to a spring of spring constant 200 N/m. If the speed of the block is 40 m/s when the displacement from equilibrium is 3 m, what is the amplitude of the oscillations? Answer: 5m • A simple pendulum has a length L. If its period is T when it is on the surface of the Earth (gravitational acceleration g ), what is its period when it is on the surface of a planet with gravitational acceleration equal to g/4? Answer: 2T PHY 2053

  8. SHM: Example Problems • Two blocks (m = 5 kg and M = 15 kg) and a spring (k = 196 N/m) are arranged on a horizontal frictionless surface. If the smaller block begins to slip when the amplitude of the simple harmonic motion is greater than 0.5 m, what is the coefficient of static friction between the two blocks? Answer: 0.5 PHY 2053

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