Simple Harmonic Motion SHM Linear Restoring Force Ideal

Simple Harmonic Motion: SHM • Linear Restoring Force: Ideal 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. R. Field 11/7/2013 University of Florida PHY 2053 1

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! x(t) • Simple Harmonic Motion (SHM): Amplitude If x(t=0) = A then The period T is the time is takes for one circular revolution: Time t R. Field 11/7/2013 University of Florida T = period (in s) f = frequency (in Hz) PHY 2053 w = angular frequency (in rad/sec) 2

SHM: Graphical Representation If x(t=0) = A then R. Field 11/7/2013 University of Florida PHY 2053 3

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: R. Field 11/7/2013 University of Florida If x(t=0) = 0 and vx(t=0) > 0 then f = p/2: PHY 2053 4

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 R. Field 11/7/2013 University of Florida PHY 2053 5

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 R. Field 11/7/2013 University of Florida PHY 2053 (physical pendulum) 6

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: 5 m • 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: 2 T R. Field 11/7/2013 University of Florida PHY 2053 7

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 R. Field 11/7/2013 University of Florida PHY 2053 8