M3 January 2010 Q2

2. A particle \(P\) moves in a straight line with simple harmonic motion of period 2.4 s about a fixed origin \(O\). At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(P\) is at \(O\). When \(t = 0.4\), \(v = 4\). Find

(a) the greatest speed of \(P\), (7)

(b) the magnitude of the greatest acceleration of \(P\). (2)

Mark scheme (a)

Scheme	Marks
\(\dfrac{2\pi}{\omega} = 2.4 \quad\Rightarrow \omega = \dfrac{5\pi}{6}\ (\approx 2.62)\)	M1 A1
\(x = 0,\ t = 0 \quad\Rightarrow\quad x = a\sin\omega t\)
when \(t = 0.4,\ \ x = a\sin\left(\dfrac{5\pi}{6} \times 0.4\right) \qquad \left(= \dfrac{\sqrt{3}}{2}a\right)\)	M1
\(v^2 = \omega^2\left(a^2 - x^2\right) \quad\Rightarrow\quad 16 = \dfrac{25\pi^2}{36}\left(a^2 - \dfrac{3a^2}{4}\right) \quad\Rightarrow\quad a = \dfrac{48}{5\pi}\ (\approx 3.06)\)	M1 A1
\(v_{\max} = a\omega = 8\) (or awrt 8.0 if decimals used earlier) cao	M1 A1
	(7)

Alternative in (a)

\(\dfrac{2\pi}{\omega} = 2.4 \Rightarrow \omega = \dfrac{5\pi}{6}\)	M1 A1
\(x = 0,\ t = 0 \quad\Rightarrow\quad x = a\sin\omega t\)
\(\dot{x} = a\omega\cos\omega t\)	M1
\(4 = a\omega\cos\left(\dfrac{5\pi}{6} \times 0.4\right)\)	M1
\(a = \dfrac{48}{5\pi}\ (\approx 3.06)\quad\) or \(\quad a\omega = 8\)	A1
\(v_{\max} = a\omega = 8\)	M1 A1
	(7)

Scheme	Marks
\(\ddot{x}_{\max} = a\omega^2 = \dfrac{20\pi}{3}\) awrt 21	M1 A1
	(2)
	(9 marks)