Page 315 - Physics Coursebook 2015 (A level)

Chapter 19: Oscillations
Summary
■■ Many systems, mechanical and otherwise, will oscillate freely when disturbed from their equilibrium position.
■■ Some oscillators have motion described as simple harmonic motion (s.h.m.). For these systems, graphs of displacement, velocity and acceleration against time are sinusoidal curves – see Figure 19.37.
■■ A body executes simple harmonic motion if its acceleration is directly proportional to its displacement from its equilibrium position, and is always directed towards the equilibrium position.
■■ Acceleration a in s.h.m. is related to displacement x by the equation a = −ω2x.
■■ The maximum speed v0 in s.h.m. is given by the equation:
v0 = ωx0.
■■ The frequency or period of a simple harmonic oscillator
is independent of its amplitude.
■■ In s.h.m., there is a regular interchange between kinetic energy and potential energy.
■■ Resistive forces remove energy from an oscillating system. This is known as damping. Damping causes the amplitude to decay with time.
■■ Critical damping is the minimum amount of damping required to return an oscillator to its equilibrium position without oscillating.
■■ When an oscillating system is forced to vibrate close
to its natural frequency, the amplitude of vibration increases rapidly. The amplitude is maximum when the forcing frequency matches the natural frequency of the system; this is resonance.
■■ Resonance can be a problem, but it can also be very useful.
0
0
0
Time
Time
Time
Figure 19.37 Graphs for s.h.m.
■■ During a single cycle of s.h.m., the phase changes by 2π radians. The angular frequency ω of the motion is related to its period T and frequency f:
ω = 2π and ω = 2πf T
■■ In s.h.m., displacement x and velocity v can be represented as functions of time t by equations of the form:
x = x0sinωt and v = v0cosωt
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Acceleration, a Velocity, v Displacement, x