Edexcel A-Level Mathematics A2 M3.3.2 Simple Harmonic Motion Questions

Practise deriving simple harmonic motion from force equations, finding angular frequency and amplitude, calculating speeds at displacements, and using sine or cosine to obtain exact times.

Exam points

Show SHM by deriving ẍ=-ω²x from the particle’s force equation for the model.
Use v²=ω²(a²-x²) to find speed at a named displacement from equilibrium.
Use sine or cosine displacement formulae to find exact times within one oscillation.

[Maximum number: 7]

The fixed point A is vertically above the fixed point B, with A B=3 l

A light elastic string has natural length l and modulus of elasticity 4 m g One end of the string is attached to A and the other end is attached to a particle P of mass m

A second light elastic string also has natural length l and modulus of elasticity 4 m g One end of this string is attached to P and the other end is attached to B.

Initially P rests in equilibrium at the point E, where AEB is a vertical straight line.

(a)

Show that $\ddot{x}=-\frac{8 g}{l} x$

[ 4 ]

Top string tension = $4mg\left(\frac{5}{8}+\frac{x}{l}\right)$ , bottom string tension = $4mg\left(\frac{3}{8}-\frac{x}{l}\right)$

mg+4mg\left(\frac{3}{8}-\frac{x}{l}\right)-4mg\left(\frac{5}{8}+\frac{x}{l}\right)=m\ddot{x}

m\ddot{x}=-\frac{8mg}{l}x

A1*

\ddot{x}=-\frac{8g}{l}x

(b)

Find the length of time, in each complete oscillation, for which P is more than 1.5 l from A, giving your answer in terms of g and l

[ 3 ]

$t=\sqrt{\frac{l}{2g}}\cos^{-1}\left(-\frac{1}{3}\right)$

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Edexcel A-Level Mathematics A2 M3.3.2 Simple Harmonic Motion Questions question 1

Question (a)

Question (b)