Simple Harmonic Motion (Part 3) – A2 Physics Revision

Have you read parts 1 and 2 of this post?

Spring Oscillator Period and Frequency

  • Since a = -4π2f2A sin(2πft) and s = A sin(2πft), you could say a = -4π2f2s
  • Equating Newton’s Second Law and Hooke’s Law, you get ma = -kx
  • Substituting in the expression for a above, cancelling the displacement with the extension of the spring (as they’re the same thing) and rearranging for F (which you probably won’t have to do, barring a nasty “show that” question), you get:
  • f = 1/(2π) √(m/k)
  • Or, since frequency and period are reciprocals:
  • T = 2π √(k/m)
  • These equations show how the period and frequency of a spring oscillator depend on the spring constant of the spring and the mass of the object
  • Remember: The frequency and period don’t change as the amplitude of the oscillations decreases – they’re independent of amplitude
  • The mass and spring constant affect the period of a spring oscillator

Pendulum Period and Frequency

  • Similarly, equating F = (mgs)/L with F = ma and substituting in a = -4π2f2s, you can obtain equations for the period and frequency of a pendulum:
  • f = 1/(2π) √(g/L)
  • T = 2π √(L/g)
  • The period and frequency of a pendulum are affected by the length of the string and the gravitational field strength / acceleration due to gravity
  • They’re not affected by the amplitude
  • For spring oscillators, the mass affects the period, but for pendulums it does not
  • Pendulums are used in old clocks because even when they lose energy due to air resistance and their amplitude decreases, their period is unaffected

Energy Conversions

  • Simple harmonic oscillators are faster in the centre than they are at the extremes of their oscillations (as shown by the velocity graph and the displacement graph)
  • Kinetic Energy is 1/2 mv2, so they have the most kinetic energy when they’re fastest and zero kinetic energy at the ends, where they stop for an instant
  • The kinetic energy is therefore in phase with the velocity, i.e. 90° degrees out of phase with displacement and 90° degrees out of phase with acceleration
  • Where does that kinetic energy go when it decreases to zero? It’s temporarily converted into another form of energy
  • In pendulums, it’s the gravitational potential energy they gain by swinging out to the side and increasing their height
  • In horizontal spring oscillators, it’s elastic potential energy from the compression or extension of the spring
  • In vertical spring oscillators, it’s a combination of elastic potential energy and gravitational potential energy
  • The total energy in the system (potential energy added to kinetic energy) is always the same
  • The potential energy is 90° out of phase with the kinetic energy
  • In case you need to actually add them up, Elastic Potential Energy is E = 1/2 kx2, Kinetic Energy is E = 1/2 mv2 and Gravitational Potential Energy is E = mgΔh
  • The total energy is the sum of those, or, for a spring oscillator, E = 1/2 kA2 where A is the amplitude and k is the spring constant

Energy Loss

  • Technically, the energy in the system does decrease, because the mass will usually be oscillating through something such as air
  • The mass collides with air particles and loses tiny amounts of kinetic energy. Eventually, this adds up
  • This is why the amplitude of pendulums gradually decreases – they are losing energy
  • The amplitude of the oscillator is proportional to the total energy in the system
  • The oscillator is said to be damped, as there is something taking energy from it
  • The damping effect would be greater if the oscillator were in a more dense medium, such as water
  • The energy decreases by a fixed percentage each oscillation
  • When its amplitude is greatest, a pendulum loses the most energy each swing, but when the pendulum is only swinging a tiny amount, it loses a much smaller amount of energy each swing
  • The rate of loss of energy is proportional to the amount of energy present
  • Does that sound familiar? The energy loss is exponential!
  • The loss of energy from damped oscillators follows exponential decay
  • Remember that the period and frequency are unaffected by the loss of energy / amplitude

Resonance

  • Similarly to how damping removes energy from an oscillating system, energy can be added to the system from outside
  • If the energy is applied at the right time, to coincide with the oscillations, the total energy in the system (and thus the amplitude) can be increased
  • Pushing a swing is the best analogy for this – you push when it’s going away from you, not when it’s coming towards you
  • Objects have a natural frequency which depends on various physical properties
  • If energy is applied to an object at its natural frequency, it oscillates and its amplitude increases dramatically
  • This can be a bad thing (bridges can be accidentally destroyed by marching on them, causing them to resonate), so damping is often used to counteract the effects of resonance
  • Like I said, I don’t have time to draw any more graphs, so you’ll have to imagine this next one (or just look it up in your notes / proper revision guide)
  • A graph with frequency on the x-axis and amplitude on the y-axis shows how the frequency external energy is applied to an object at affects the amplitude the object is made to oscillate at
  • The graph will have a sharp peak at the object’s natural frequency, and will curve towards zero on either side of the peak
  • If the object has been damped to reduce the effects of resonance, the peak will be flatter and wider (with a smaller maximum amplitude)

Data, Formulae and Relationships Booklet

  • T = 2π √(m/k) is in the formula booklet, so you can work out the period of a spring oscillator
  • If you want to derive it, F = kx is in the formula book (without the minus sign), there’s no way you could forget F = ma and a = -4π2f2s is in the formula booklet, disguised as a = -(2πf)2x, hidden in a larger expression at the start of the Oscillations section
  • f = 1/T is in the formula booklet, so you can work out the frequency from the period
  • The equations for a pendulum’s period and frequency are missing, as is the F = mgs/L approximation used to derive them, but I’m not sure whether you actually need those equations
  • The formula for the total energy in a spring oscillator is given and it’s also broken down into kinetic energy and elastic potential energy (but gravitational potential energy is ignored, so this must be a horizontal oscillator)
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About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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