What does the minus sign in \ddot x=-\omega^2x mean?

The acceleration is directed towards the centre, opposing the displacement.

x=a\sin\omega t means where is the particle at t=0?

At the centre, x=0, moving in the positive direction

How do you prove motion is SHM?

Apply Newton's second law and show \ddot x=-\omega^2x, with x measured from the centre.

How do you find a time between two points?

Solve the trig equation for each point separately (first occurrence), then subtract.

Simple harmonic motionEdexcel International A Level Further Maths: Flashcards

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Defining equation of SHM?

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Defining equation of SHM?: $\ddot x=-\omega^2x$
What does the minus sign in $\ddot x=-\omega^2x$ mean?: The acceleration is directed towards the centre, opposing the displacement.
Period of SHM?: $T=\frac{2\pi}{\omega}$
Formula for speed at displacement $x$ ?: $v^2=\omega^2(a^2-x^2)$
Maximum speed in SHM, and where?: $a\omega$ , at the centre $O$
Maximum acceleration in SHM, and where?: $a\omega^2$ , at the ends $x=\pm a$
$x=a\cos\omega t$ means where is the particle at $t=0$ ?: At an extreme position, $x=a$
$x=a\sin\omega t$ means where is the particle at $t=0$ ?: At the centre, $x=0$ , moving in the positive direction
Frequency in terms of $\omega$ ?: $f=\frac{\omega}{2\pi}$ hertz
How do you prove motion is SHM?: Apply Newton's second law and show $\ddot x=-\omega^2x$ , with $x$ measured from the centre.
Speed at the ends of the motion?: Zero
Acceleration at the centre?: Zero
How do you find a time between two points?: Solve the trig equation for each point separately (first occurrence), then subtract.

Exam questions on Simple harmonic motion

A particle moves in a straight line with simple harmonic motion about a centre $O$ . The amplitude of the motion is $0.5$ m and the period is $2$ s.
Find the speed of the particle when it is $0.3$ m from $O$ .2 marks
The displacement of a particle $P$ from a fixed point $O$ on a straight line, at time $t$ seconds, is $x=0.4\cos3t$ metres.
Find the first time at which $P$ is at $x=-0.2$ m.2 marks
A particle $P$ of mass $2$ kg moves on a smooth horizontal straight line. When $P$ is at displacement $x$ metres from a fixed point $O$ on the line, the only horizontal force on $P$ is directed towards $O$ and has magnitude $18x$ N.
Show that $P$ moves with simple harmonic motion, and find the period of the motion.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).