M3 June 2006 Q7
7. One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(m\). The other end is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical when it is projected horizontally with speed \(\sqrt{\dfrac{5gl}{2}}\).
(a) Find the speed of \(P\) when the string is horizontal. (4)
When the string is horizontal it comes into contact with a small smooth fixed peg which is at the point \(B\), where \(AB\) is horizontal, and \(AB < l\). Given that the particle then describes a complete semicircle with centre \(B\),
(b) find the least possible value of the length \(AB\). (9)
| Scheme | Marks |
|---|---|
| Conservation of Energy | |
| \(\dfrac{1}{2}m\left(\dfrac{5gl}{2} - u^2\right) = mgl\) | M1 A1= A1 |
| Leading to \(u = \sqrt{\left(\dfrac{gl}{2}\right)}\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| Conservation of Energy | |
| \(\dfrac{1}{2}m\left(u^2 - v^2\right) = mgr\) | M1 A1 |
| \(v^2 = u^2 - 2gr\) | |
| \(R(\downarrow)\) \(T + mg = \dfrac{mv^2}{r}\) | M1 A1 |
| \(T = \dfrac{m}{r}\left(u^2 - 2gr\right) - mg\) | M1 |
| \(= \dfrac{mu^2}{r} - 3mg\) | A1 |
| \(= \dfrac{mgl}{2r} - 3mg\) | M1 |
| \(T \geqslant 0 \Rightarrow \dfrac{mgl}{2r} \geqslant 3mg\) | M1 |
| \(\Rightarrow \dfrac{l}{6} \geqslant r\) | |
| \(AB_{\text{MIN}} = \dfrac{5l}{6}\) | A1 |
| (9) | |
| (13 marks) |
Notes
The scheme brackets the M1 marks: the later M1 marks depend on the energy and N2L M1 marks.
(Corrected from the printed mark scheme: \(\dfrac{l}{6} \geqslant r\) is printed as \(\dfrac{1}{6} \geqslant r\).)