Maximum acceleration in SHM, and where?

a\omega^2 at the ends of the motion.

Two forms of the displacement?

x=a\sin\omega t from the centre, x=a\cos\omega t from an end.

Why does the weight not change the period of a hanging mass?

The extra extension at equilibrium cancels the weight, so the resultant is -\frac{\lambda x}{l}.

When does an elastic string stop giving SHM?

When the string goes slack, i.e. the amplitude exceeds the equilibrium extension.

How do you prove motion is SHM?

Show the resultant force at displacement x gives \ddot{x}=-\omega^2x.

Simple harmonic motionEdexcel A-Level Further Maths: Flashcards

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Equation defining SHM?

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Equation defining SHM?: $\ddot{x}=-\omega^2x$ .
Period of SHM?: $T=\frac{2\pi}{\omega}$ .
Speed at displacement $x$ ?: $v^2=\omega^2(a^2-x^2)$ .
Maximum speed in SHM, and where?: $a\omega$ at the centre.
Maximum acceleration in SHM, and where?: $a\omega^2$ at the ends of the motion.
Two forms of the displacement?: $x=a\sin\omega t$ from the centre, $x=a\cos\omega t$ from an end.
Hooke's law for tension?: $T=\frac{\lambda e}{l}$ .
Elastic potential energy stored?: $\frac{\lambda e^2}{2l}$ .
Vertical spring: expression for $\omega^2$ ?: $\omega^2=\frac{\lambda}{ml}$ .
Why does the weight not change the period of a hanging mass?: The extra extension at equilibrium cancels the weight, so the resultant is $-\frac{\lambda x}{l}$ .
When does an elastic string stop giving SHM?: When the string goes slack, i.e. the amplitude exceeds the equilibrium extension.
How do you prove motion is SHM?: Show the resultant force at displacement $x$ gives $\ddot{x}=-\omega^2x$ .

Exam questions on Simple harmonic motion

A particle $P$ moves in a straight line with simple harmonic motion about a fixed centre $O$ . The amplitude of the motion is 0.8 m and the angular frequency is $\omega=5$ rad s⁻¹.
Given that $x=0.8\sin5t$ with $t=0$ at $O$ , find the time taken for $P$ to move directly from $O$ to a point 0.4 m from $O$ .2 marks
A particle $P$ of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.4 m and modulus of elasticity 20 N. The other end of the spring is fixed to a ceiling and $P$ hangs in equilibrium. $P$ is then pulled vertically downwards a distance 0.05 m from the equilibrium position and released from rest. Take $g=9.8$ m s⁻².
Find the maximum speed of $P$ .2 marks
A particle $P$ of mass 0.4 kg lies on a smooth horizontal table. It is attached to two identical light elastic strings, each of natural length 0.8 m and modulus of elasticity 16 N. The other ends of the strings are fixed to points $A$ and $B$ on the table, where $AB=2.4$ m, and $P$ rests in equilibrium at the midpoint $M$ of $AB$ . $P$ is then displaced along $AB$ and released from rest.
$P$ is displaced a distance $x$ metres from $M$ towards $B$ , with both strings taut. Show that $P$ moves with simple harmonic motion, and state the value of $\omega$ .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).