Vibrations and Waves AP Physics Lecture Notes m Vibrations and Waves

Units of Chapter 11 Simple Harmonic Motion Energy in the Simple Harmonic Oscillator The Period and Sinusoidal Nature of SHM The Simple Pendulum Vibrations and Waves

Simple Harmonic Motion If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic (T). m We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 ). X = 0 Vibrations and Waves

Simple Harmonic Motion m x = 0 The minus sign on the force indicates that it is a restoring force – it is directed to restore the mass to its equilibrium position. The force exerted by the spring depends on the displacement: m x F Vibrations and Waves

Simple Harmonic Motion m x F (a) (k) is the spring constant (b) Displacement (x) is measured from the equilibrium point (c) Amplitude (A) is the maximum displacement (e) Period (T) is the time required to complete one cycle (f) Frequency (f) is the number of cycles completed per second (d) A cycle is a full to-and-fro motion Vibrations and Waves

Simple Harmonic Motion If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force. m xoxo mg Equilibrium Position Vibrations and Waves

11-1 Simple Harmonic Motion Any vibrating system where the restoring force is proportional to the negative of the displacement moves with simple harmonic motion (SHM), and is often called a simple harmonic oscillator. Vibrations and Waves

Energy in the Simple Harmonic Oscillator Potential energy of a spring is given by: The total mechanical energy is then: The total mechanical energy will be conserved Vibrations and Waves

Energy in the Simple Harmonic Oscillator If the mass is at the limits of its motion, the energy is all potential. m A m x = 0 v ma x If the mass is at the equilibrium point, the energy is all kinetic. Vibrations and Waves

Energy in the Simple Harmonic Oscillator This can be solved for the velocity as a function of position: The total energy is, therefore And we can write: where Vibrations and Waves

The Period and Sinusoidal Nature of SHM If we look at the projection onto the x axis of an object moving in a circle of radius A at a constant speed v max, we find that the x component of its velocity varies as: This is identical to SHM. A v ma x  x v Vibrations and Waves

The Period and Sinusoidal Nature of SHM Therefore, we can use the period and frequency of a particle moving in a circle to find the period and frequency: Vibrations and Waves

The Period and Sinusoidal Nature of SHM The acceleration can be calculated as function of displacement m x F Vibrations and Waves

The Simple Pendulum A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible. Vibrations and Waves

mg  F s x Small angles x  s k for SHM Simple Pendulum L m  Vibrations and Waves