Edexcel IAL Mathematics M3.3.2 simple harmonic motion

Practise SHM by deriving the motion equation, identifying ω and amplitude, then finding speeds, periods and time spent in intervals.

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 ]

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M3.3.2 - Simple harmonic motion question 1

Question (a)

Question (b)