AQA A-Level Physics/Simple Harmonic Motion
Simple Harmonic Motion- Objects can oscillate in all sorts of ways but a really important form of oscillations is SHM or Simple Harmonic Motion. An object is undergoing SHM if:
- The acceleration of the object is directly proportional to its displacement from its equilibrium position.
- The acceleration is always directed towards the equilibrium position.
The frequency of an oscillation is measured in Hertz, and is the number of oscillations per second. The time for one oscillation is known as the period (T) and is measured in seconds.
Acceleration: we can calculate the acceleration of the object at any point in its oscillation by using this equation.
In this equation; a is the acceleration, f is the frequency in Hertz and x is displacement from the central position in metres.
Displacement: (When using this equation in the calculator, make sure that it’s in radians) You can calculate the displacement of the object at any point in its oscillation using this equation.
The terms of this equation are the same as that of acceleration. The extra terms in the equation are A which is the amplitude (or maximum displacement) in metres, t the time since the oscillation began in seconds.
Velocity: We can calculate the velocity of the object at any point in its oscillation using the equation below
The terms in this equation are the same as the equations above. The extra term in this equation is v, which funnily enough is the velocity.
These graphs show displacement, velocity and acceleration respectively. These graphs show how displacement and acceleration are proportional but in opposite directions, and also how when you have the minimum displacement, velocity is at its maximum. These are important notions to remember as they’re specific to SHM and allow us to determine whether an object or system does in-fact move with SHM.
The velocity equation simplifies to the equation below when trying to find the maximum speed (which will be at the point of minimum displacement)
The acceleration equation simplifies to the equation below when we just want to know the maximum acceleration (which is at the point of maximum displacement e.g why it uses the Amplitude of the system)
SHM and Energy: For a pendulum undergoing SHM energy is being transferred back and forth between kinetic energy and potential energy. The total energy remains the same and is equal to the kinetic energy + potential energy at any point within the motion.