Oscillations © 2014 Pearson Education, Inc.

Periodic Motion Periodic motion is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time. Amplitude A Period (seconds,s) Period, T, is the time for one complete oscillation. (seconds,s) Frequency Hertz (s -1 ) Frequency, f, is the number of complete oscillations per second. Hertz (s -1 )

Simple Harmonic Motion, SHM Simple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx xF

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 on the previous figure). Simple Harmonic Motion—Spring Oscillations

The minus sign on the force indicates that it is a restoring force—it is directed to restore the mass to its equilibrium position. k is the spring constant The force is not constant, so the acceleration is not constant either

Simple Harmonic Motion—Spring Oscillations summary Displacement is measured from the equilibrium point Amplitude is the maximum displacement, A A cycle is a full to-and-fro motion; this figure shows half a cycle Period is the time required to complete one cycle, T Frequency is the number of cycles completed per second, f

Simple Harmonic Motion—Spring Oscillations 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.

Simple Harmonic Motion—Spring Oscillations Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator.

Displacement in SHM m x = 0x = +Ax = -A x Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left. The maximum displacement is called the amplitude A.

Velocity in SHM m x = 0 x = +A x = -A v (+) Velocity is positive when moving to the right and negative when moving to the left.Velocity is positive when moving to the right and negative when moving to the left. It is zero at the end points and a maximum at the midpoint in either direction (+ or -).It is zero at the end points and a maximum at the midpoint in either direction (+ or -). v (-)

Acceleration in SHM m x = 0x = +Ax = -A Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.)Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.) Acceleration is a maximum at the end points and it is zero at the center of oscillation. +x -a -x +a

Acceleration vs. Displacement m x = 0x = +Ax = -A x v a Given the spring constant, the displacement, and the mass, the acceleration can be found from: or Note: Acceleration is always opposite to displacement.

Energy in Simple Harmonic Motion We already know that the potential energy of a spring is given by: PE = ½ kx 2 The total mechanical energy is then: The total mechanical energy will be conserved, as we are assuming the system is frictionless.

Energy in Simple Harmonic Motion If the mass is at the limits of its motion, the energy is all potential. If the mass is at the equilibrium point, the energy is all kinetic. We know what the potential energy is at the turning points: (11-4a)

The total energy is, therefore ½ kA 2 And we can write: This can be solved for the velocity as a function of position: where Energy in Simple Harmonic Motion

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:

11-3 The Period and Sinusoidal Nature of SHM We can similarly find the position as a function of time: © 2014 Pearson Education, Inc.

The Period and Sinusoidal Nature of SHM The top curve is a graph of the previous equation. The bottom curve is the same, but shifted ¼ period so that it is a sine function rather than a cosine.

The Period and Sinusoidal Nature of SHM The velocity and acceleration can be calculated as functions of time; the results are below, and are plotted at left.