M3 January 2006 Q7
7.
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2a\). The other end of the string is fixed to a point \(A\) which is vertically above the point \(O\) on a smooth horizontal table. The particle \(P\) remains in contact with the surface of the table and moves in a circle with centre \(O\) and with angular speed \(\sqrt{\left(\dfrac{kg}{3a}\right)}\), where \(k\) is a constant. Throughout the motion the string remains taut and \(\angle APO = 30^\circ\), as shown in Figure 3.
(a) Show that the tension in the string is \(\dfrac{2kmg}{3}\). (3)
(b) Find, in terms of \(m\), \(g\) and \(k\), the normal reaction between \(P\) and the table. (3)
(c) Deduce the range of possible values of \(k\). (2)
The angular speed of \(P\) is changed to \(\sqrt{\left(\dfrac{2g}{a}\right)}\). The particle \(P\) now moves in a horizontal circle above the table. The centre of this circle is \(X\).
(d) Show that \(X\) is the mid-point of \(OA\). (7)
| Scheme | Marks |
|---|---|
| N2L \(\leftarrow\ \ T\cos 30^\circ = m(2a\cos 30^\circ)\left(\dfrac{kg}{3a}\right)\) | M1 A1 |
| \(T = \dfrac{2kmg}{3}\) * cso | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\uparrow\ \ R = mg - T\sin 30^\circ\) | M1 A1 |
| \(= mg\left(1 - \dfrac{k}{3}\right)\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \((R \geqslant 0) \Rightarrow k \leqslant 3\) ignore \(k > 0\), accept \(k < 3\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| N2L \(\leftarrow\ \ T\cos\theta = m(2a\cos\theta)\left(\dfrac{2g}{a}\right)\) | M1 A1 |
| \((T = 4mg)\) | |
| \(\uparrow\ \ T\sin\theta = mg\) | M1 |
| Eliminating T | M1 |
| \(AX = 2a\sin\theta = \dfrac{1}{2}a\) | A1 |
| \(AO = 2a\sin 30^\circ = a \Rightarrow AX = \dfrac{1}{2}AO\), as required * cso | B1, A1 |
| (7) | |
| (15 marks) |