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Harmonic Motion: Velocity and Acceleration - Physics 1D03 Lecture 33

This lecture discusses the concepts of velocity and acceleration in harmonic motion. It covers topics such as determining amplitude, frequency, period, position at a given time, velocity, and acceleration. Examples are provided to illustrate these concepts. The lecture also explores the conditions for simple harmonic motion and the relationship between mass and spring constant.

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Harmonic Motion: Velocity and Acceleration - Physics 1D03 Lecture 33

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  1. Harmonic Motion (III) Physics 1D03 - Lecture 33

  2. Recall: Velocity and Acceleration Physics 1D03 - Lecture 33

  3. Example An object oscillates with SHM along the x-axis. Its displacement from the origin varies with time according to the equation:x(t)=(4.0m)cos(πt+π/4) where t is in seconds and the angles in radians. • determine the amplitude • determine the frequency • determine the period • its position at t=0 sec • calculate the velocity at any time, and the vmax • calculate the acceleration at any time, and amax Physics 1D03 - Lecture 33

  4. When do we have Simple Harmonic Motion ? A system exhibits SHM is we find that acceleration is directly proportionalto displacement: a(t) = - w2 x(t) SHM is also called ‘oscillatory’ motion. Its is called ‘harmonic’ because the sine and cosine function arecalled harmonic functions, and they are solutions to the abovedifferential equation – lets prove it !!!SHM is ‘periodic’. Physics 1D03 - Lecture 33

  5. Massand Spring F = -kx Newton’s 2nd Law: M so x This is a 2nd order differential equation for the function x(t). Recall that for SHM, a = -w2 x : the above is identical except for the proportionality constant. Hence, we must have: or: Physics 1D03 - Lecture 33

  6. Example A 7.0 kg mass is hung from the bottom end of a vertical spring fastened to the ceiling. The mass is set into vertical oscillations with a period of 2.6 s. Find the spring constant (aka force constant of the spring). Physics 1D03 - Lecture 33

  7. Example A block with a mass of 200g is connected to a light spring with a spring constant k=5.0 N/m and is free to oscillate on a horizontal frictionless surface. The block is displaced 5.0cm from equilibrium and released from rest. • find the period of its motion • determine the maximum speed of the block • determine the maximum acceleration of the block Physics 1D03 - Lecture 33

  8. Example A 1.00 kg mass on a frictionless surface is attached to a horizontal spring. The spring is initially stretched by 0.10 m and the mass is released from rest. The mass moves, and after 0.50 s, the speed of the mass is zero. What is the maximum speed of the mass ??? Physics 1D03 - Lecture 33

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