# Further Maths - Simple Harmonic Motion

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/simple-harmonic-motion/ · Updated: 2022-03-10 · Tags: school

## See Also
- [Further Maths - Differential Equations](https://ollybritton.com/notes/a-level/further-maths/topics/differential-equations/)

## Flashcards
### 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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