Further Maths - Simple Harmonic Motion

Created: March 10, 2022

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2022-03-10

What, in “dot” notation, is the formula for simple harmonic motion in terms of angular velocity, $\omega$?
\[\ddot{x} = -\omega^2 x\]

When asked to prove a relationship in further maths simple harmonic motion from

\[\frac{\text{d}^2x}{\text{d}t^2} = -\omega^2 x\]

what is a useful manipulation?
\[\frac{\text{d}^2 x}{\text{d}t^2} = \frac{\text{d}v}{\text{d}t} = \frac{\text{d}v}{\text{d}x} \times \frac{\text{d}x}{\text{d}t}\]

How can you write simple harmonic motion in terms of $\ddot{x}$ and $\frac{\text{d}v}{\text{d}x}$?
\[\ddot{x} = v\frac{\text{d}v}{\text{d}x}\]

What is the general equation for damped harmonic motion?
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for heavy damping?
\[k^2 > 4\omega^2\]

What happens in heavy damping?

There are no oscillations.

What sort of equation do you get for heavy damping in simple harmonic motion?
\[Ae^{kx} + Be^{kx}\]
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for critical damping?
\[k^2 = 4\omega^2\]

What happens in critical damping?

There are no oscillations.

What sort of equation do you get for heavy damping in simple harmonic motion?
\[(A + Bx)e^{kx}\]
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for light damping?
\[k^2 > 4\omega^2\]

What happens in light damping?

The amplitude of oscillations decreases exponentially with time.

What sort of equation do you get for light damping in simple harmonic motion?
\[e^{-px}(A\cos(kx) + B\sin(kx))\]

What roots of the complementary equation make heavy damping?

Two real roots.

What roots of the complementary equation make critical damping?

Equal roots.

What roots of the complementary equation make light damping?

Complex roots.

2022-03-30

What energy stores are there in a spring and mass simple harmonic oscillator?
  • Kinetic energy
  • Potential energy
  • Elastic energy

When measuring the period or frequency of a mass on a string, how could you reduce the effects of air resistance?

Record one oscillation accurately using a timer.

What do you need to use to work with angles in a practical?

A protractor.

2022-05-11

In SHM, is velocity proportional to displacement?

No.

What must be specified about a graph showing proportionality of $a$ and $x$ for it to be simple harmonic motion?

It is a straight line passing through the origin.



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