Further Maths - Simple Harmonic Motion

Pearson Edexcel Further Mathematics 2022

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

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The SHM equation and useful manipulations

What, in “dot” notation, is the formula for simple harmonic motion in terms of angular velocity, $\omega$?

shm-equation-dot-notation
\[\ddot{x} = -\omega^2 x\]

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?

shm-acceleration-chain-rule
\[\frac{\text{d}^2 x}{\text{d}t^2} = \frac{\text{d}v}{\text{d}t} = \frac{\text{d}v}{\text{d}x} \times \frac{\text{d}x}{\text{d}t}\]

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

shm-acceleration-v-dvdx
\[\ddot{x} = v\frac{\text{d}v}{\text{d}x}\]

Damped harmonic motion

What is the general equation for damped harmonic motion?

damped-harmonic-motion-equation
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for _ _ heavy damping _ _ ?

heavy-damping-condition
\[k^2 > 4\omega^2\]

What happens in heavy damping?

heavy-damping-no-oscillations

There are no oscillations.

What sort of equation do you get for heavy damping in simple harmonic motion?

heavy-damping-general-solution-form
\[Ae^{kx} + Be^{kx}\]
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for _ _ critical damping _ _ ?

critical-damping-condition
\[k^2 = 4\omega^2\]

What happens in critical damping?

critical-damping-no-oscillations

There are no oscillations.

What sort of equation do you get for heavy damping in simple harmonic motion?

critical-damping-general-solution-form
\[(A + Bx)e^{kx}\]
\[\frac{\text{d}^2x}{\text{d}t^2} + k \frac{\text{d}x}{\text{d}t} + \omega^2 x = 0\]

What are the conditions for _ _ light damping _ _ ?

light-damping-condition
\[k^2 < 4\omega^2\]

What happens in light damping?

light-damping-exponential-decay

The amplitude of oscillations decreases exponentially with time.

What sort of equation do you get for light damping in simple harmonic motion?

light-damping-general-solution-form
\[e^{-px}(A\cos(kx) + B\sin(kx))\]

What roots of the complementary equation make heavy damping?

heavy-damping-two-real-roots

Two real roots.

What roots of the complementary equation make critical damping?

critical-damping-equal-roots

Equal roots.

What roots of the complementary equation make light damping?

light-damping-complex-roots

Complex roots.

Practical and properties of SHM

What energy stores are there in a spring and mass simple harmonic oscillator?

shm-spring-mass-energy-stores
  • 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?

shm-period-measurement-air-resistance

Record one oscillation accurately using a timer.

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

shm-practical-protractor-for-angles

A protractor.

In SHM, is velocity proportional to displacement?

shm-velocity-not-proportional-to-displacement

No.

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

shm-acceleration-displacement-graph-condition

It is a straight line passing through the origin.



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