M3 June 2006 Q3
3. A particle \(P\) of mass 0.2 kg oscillates with simple harmonic motion between the points \(A\) and \(B\), coming to rest at both points. The distance \(AB\) is 0.2 m, and \(P\) completes 5 oscillations every second.
(a) Find, to 3 significant figures, the maximum resultant force exerted on \(P\). (6)
When the particle is at \(A\), it is struck a blow in the direction \(BA\). The particle now oscillates with simple harmonic motion with the same frequency as previously but twice the amplitude.
(b) Find, to 3 significant figures, the speed of the particle immediately after it has been struck. (5)
| Scheme | Marks |
|---|---|
| \(a = 0.1\) | B1 |
| \(\dfrac{2\pi}{\omega} = \dfrac{1}{5} \Rightarrow \omega = 10\pi\) | M1 A1 |
| \(F_{\max} = ma\omega^2\) | M1 |
| \(= 0.2 \times 0.1 \times (10\pi)^2\) | M1 |
| \(\approx 19.7\) (N) cao | A1 |
| (6) |
Notes
The scheme brackets the two M1 marks for \(F_{\max}\): the second depends on the first.
| Scheme | Marks |
|---|---|
| \(a' = 0.2, \quad \omega' = 10\pi\) | B1ft, B1ft |
| \(v^2 = \omega^2\left(a^2 - x^2\right) = 100\pi^2\left(0.2^2 - 0.1^2\right)\ \ \left(= 3\pi^2 \approx 29.6\ldots\right)\) | M1 A1 |
| \(v \approx 5.44\) (m s\(^{-1}\)) cao | A1 |
| (5) | |
| (11 marks) |
Notes
If answers are given to more than 3 significant figures a maximum of one A mark is lost in the question.