M3 June 2007 Q5

5. A particle \(P\) moves on the \(x\)-axis with simple harmonic motion about the origin \(O\) as centre. When \(P\) is a distance 0.04 m from \(O\), its speed is 0.2 m s\(^{-1}\) and the magnitude of its acceleration is 1 m s\(^{-2}\).

(a) Find the period of the motion. (3)

The amplitude of the motion is \(a\) metres.

Find

(b) the value of \(a\), (3)

(c) the total time, within one complete oscillation, for which the distance \(OP\) is greater than \(\dfrac{1}{2}a\) metres. (5)

Scheme	Marks
\(\ddot{x} = -\omega^2 x \Rightarrow 1 = \omega^2 \times 0.04\ \ (\Rightarrow \omega = 5)\)	M1 A1
\(T = \dfrac{2\pi}{5}\) awrt 1.3	A1
	(3)

Mark scheme (b)

Scheme	Marks
\(v^2 = \omega^2\left(a^2 - x^2\right) \Rightarrow 0.2^2 = 5^2\left(a^2 - 0.04^2\right)\) ft their \(\omega\)	M1 A1ft
\(a = \dfrac{\sqrt{2}}{25}\) accept exact equivalents or awrt 0.057	A1
	(3)

Mark scheme (c)

Scheme	Marks
Using \(x = a\cos\omega t\)
\(\dfrac{1}{2}a = a\cos\omega t\) ft their \(\omega\)	M1 A1ft
\(5t = \dfrac{\pi}{3}\)
\(t = \dfrac{\pi}{15}\)	A1
\(T' = 4t = \dfrac{4\pi}{15}\) awrt 0.84	M1 A1
	(5)
	(11 marks)

Alternative to (c)

Using \(x = a\sin\omega t\)
\(\dfrac{1}{2}a = a\sin\omega t\) ft their \(\omega\)	M1 A1ft
\(5t = \dfrac{\pi}{6}\)
\(t = \dfrac{\pi}{30}\)	A1
\(T' = T - 4t = \dfrac{4\pi}{15}\) awrt 0.84	M1 A1 (5)