Chapter 5 Oscillatory Motion
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1 Chapter 5 Oscillatory Motion Simple Harmonic Motion An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely directed. Sample: A mass oscillating vertically on a spring has a pen attached to it. While the mas is oscillating, a sheet of paper is moved perpendicular to the direction of motion of the spring, and the pen traces out a wavelike pattern.
2 Example 1: When four students with a total mass of 40 kg step in their 100 kg car, the four springs of the car compress.0 cm. If the total weight of the students and the car is evenly supported by the springs, find the frequency of vibration when the car hits a bump. Calculate the time for one complete oscillation
3 The phase constant ф depends on the initial displacement and velocity of the body at t = 0. If the particle starts at x Acos t i) x = + A then ф = 0 ii) x = - A then ф = π iii) x = 0 moving in x, then ф = π/ iv) x = 0 moving in + x, then ф = 3π/ or - π/ Taking the derivatives on A t v dx dt Asin t x cos, Maximum speed of a particle moving in simple harmonic motion : v A max a dv dt Acos t Maximum acceleration of a particle moving in simple harmonic motion : a A max
4 For an object that oscillating in SHM with angular frequency ω, with given initial position(xo) and intial velocity vo at t=0, the phase constant (ф) and the amplitude (A) can be determined by : (a) Find the period and frequency. (b) Find the position and velocity at t = 0. (c) Find the maximum speed and acceleration (d) Find the velocity and acceleration at t =.0 s. Example : The piston in an engine oscillates with SHM so that its displacement varies according to the expression y = (10.0 cm) cos (t + / 6) where y is cm and t is in s. At t = 0, find (a) the displacement of the particle, (b) its velocity and (c) its acceleration, (d) find the period and amplitude of the motion Example : An object moves along the y axis in SHM starts from equilibrium position at the origin at t = 0 and moves upward (positive y). The amplitude of its motion is.00 cm and the frequency is 1.50 Hz. (a) Write down the equation of its displacement as a function of time. (b) Determine the maximum speed and maximum acceleration Energy of the Simple Harmonic Oscillator Consider the block-spring system. The spring force is the only horizontal force on the block and the vertical forces do no work, so the total mechanical energy of the system is conserved.
5 The kinetic energy, K 1 1 K mv m A sin t The Potential Energy, U 1 1 U kx ka cos t E = K + U Because = k / m From the identity sin + cos = 1 Total mechanical energy of a simple harmonic oscillator : E 1 ka While the block oscillate between A and -A, the energy is transformed between the potential energy and kinetic energy. When the block is momentarily at rest at x = A, the potential energy is equal to the total mechanical energy. At x =0, when there is no potential energy stored since the spring is not being compressed or stretched, the kinetic energy is the maximum and equal to total mechanical energy. Example : An object of mass kg is hung from a light, vertical spring of force constant 0.0 N/m. The spring is then stretched 3.00 cm from equilibrium and released. (a) Find the total energy of the system. (b) Find the maximum speed and minimum velocity of the mass. (c) Find the velocity of the mass when the displacement is.00 cm. (d) Determine the position of the mass when the speed is m/s.
6 Example : A block of mass m is attached to a spring with spring constant 8.00 N/m and undergoes SHM with an amplitude 10.0 cm. When the mass is halfway between its equilibrium position and end point, its speed is + 0 cm/s. Calculate (a) the mass, m (b) period of the motion and (c) them maximum acceleration. The Pendulum A simple pendulum consists of a point mass m, called the pendulum bob, suspended by a massless, unstretchable string of length L, as shown in figure. When the mass is displaced from equilibrium by an angle, which is measured from the vertical axis and released, it oscillates about the equilibrium position = 0. Gravitational force is the only conservative force does work on the mass and tension does no work, so the total mechanical energy is conserved. Ignoring the friction, the motion repeats itself forever. The tangential component of gravitational force F t mgsin due to the angle θ is small, sin θ Thus, Ft mg (SHM) Ex : A simple pendulum 1.00 m long is at a location where g = 9.80 m/s. At t = 0 the pendulum bob is released at an angle of 8.0. (a) Determine the period and angular frequency. (b) (b) Determine the angular position of the pendulum at t = 1.0 s.
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