Further Maths - Simple Harmonic Motion

Created: March 10, 2022 | Read markdown | About these notes | View in context | Study these flashcards |

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2022-03-10

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

\[\frac{\text{d}^2 x}{\text{d}t^2} = \frac{\text{d}v}{\text{d}t} = \frac{\text{d}v}{\text{d}x} \times \frac{\text{d}x}{\text{d}t}\]

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?
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\[\frac{\text{d}^2 x}{\text{d}t^2} = \frac{\text{d}v}{\text{d}t} = \frac{\text{d}v}{\text{d}x} \times \frac{\text{d}x}{\text{d}t}\]

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

#####

\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for heavy damping??

\[k^2 > 4\omega^2\]

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

\(\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\) What are the conditions for _ _ heavy damping _ _ ?
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\[k^2 > 4\omega^2\]

#####

\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for critical damping??

\[k^2 = 4\omega^2\]

What happens in heavy damping?
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There are no oscillations.

What sort of equation do you get for heavy damping in simple harmonic motion?
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\[Ae^{kx} + Be^{kx}\]

#####

\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for light damping??

\[k^2 > 4\omega^2\]

\(\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\) What are the conditions for _ _ critical damping _ _ ?
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\[k^2 = 4\omega^2\]

What happens in critical damping?
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There are no oscillations.

What sort of equation do you get for heavy damping in simple harmonic motion?
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\[(A + Bx)e^{kx}\]

\(\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\) What are the conditions for _ _ light damping _ _ ?
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\[k^2 > 4\omega^2\]

What happens in light damping?
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The amplitude of oscillations decreases exponentially with time.

2022-03-30

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

What roots of the complementary equation make heavy damping?
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Two real roots.

What roots of the complementary equation make critical damping?
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Equal roots.

2022-05-11

What roots of the complementary equation make light damping?
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Complex roots.

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


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