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Unformatted text preview: Chapter 8 Oscillations 8.1 Periodic Motion 8.2 Simple Harmonic Motion (SHM) 8.3 Position, Velocity and Acceleration in SHM 8.4 Simple Pendulum 8.5 Energy of Simple Harmonic Motion 8.6 Damped Harmonic Motion and Forced Vibration 8.1 Periodic Motion A motion that repeats itself over and over is referred to as periodic motion. The simplest form of periodic motion is represented by an object oscillating at the end of a uniform coil spring. Below are some useful quantities for describing periodic motion: • Period T It's the time required for one complete cycle (i.e. a complete repetition of the motion). SI Unit of period: second (s) • Frequency f It's the number of complete cycles per unit time. SI unit of frequency: Hertz (Hz) = cycle/s Period and frequency are reciprocals of one another, i.e. f T / 1 = . • Amplitude A When an object undergoes periodic motion, there is point where no net force is acting on the object. This point is called the equilibrium position. The amplitude is defined as the magnitude of the maximum displacement of the object from its equilibrium position. SI unit of amplitude: meter (m) • Angular velocity ϖ It's defined as the rate of change of angular position θ per unit time. SI unit of angular velocity: rad/s For an object rotating with angular velocity ϖ , the angular displacement t ϖ θ = ∆ . Since 1 period ↔ 2 π , f π ϖ 2 = or T / 2 π ϖ = (In words, the angular displacement is 2 π f radian per second) 1 8.2 Simple Harmonic Motion (SHM) Below figure shows the oscillating cycle of a block vibrating at the end of a uniform spring on a horizontal surface. We assume that the spring has negligible mass and the surface is frictionless. Equilibrium position x = x = A v = F max t = T t = 0 x = x = A v = F max x = F = t = T /4 x = v max F = x =  A t = T /2 x = v = F max x = t = 3 T /4 v max F = Frictionless surface 2 Experiments reveal that, for most of the case, the restoring force acting on the vibrating object follows the Hooke’s law which states that: the restoring force F exerted by a spring (with spring constant k ) is proportional to the displacement from its equilibrium position: F =  kx. (Recall that the equilibrium position is defined as the point where the net force acting on the vibrating mass is zero. For the above example, x = 0 is the equilibrium position.) The negative sign appears in the force equation, which shows that the restoring force and the displacement vector from the equilibrium position are in opposite direction. A vibrating object is called a simple harmonic oscillator if a restoring force of the form F =  kx acts on it. Its motion is called simple harmonic motion (SHM). 8.3 Position, Velocity and Acceleration in SHM Here, we compare SHM with the motion of a particle moving in a circle with constant speed, as shown in the figure. Let the projection velocity of the particle on CD as v which is a component of the tangential velocity v T of the particle. We label the radius of the circle, i.e. of the particle....
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 Fall '10
 drtong
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