Edexcel M3 (Mechanics 3) 2002 June

1. A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\) with period 2 s . At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0\), \(v = 0\) and \(P\) is at a point \(A\) where \(O A = 0.25 \mathrm {~m}\).

Find the smallest positive value of \(t\) for which \(A P = 0.375 \mathrm {~m}\). \section*{2.} \begin{figure}[h] \end{figure} A metal ball \(B\) of mass \(m\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The ball \(B\) moves in a horizontal circle with centre \(O\) vertically below \(A\), as shown in Fig. 1. The string makes a constant angle \(\alpha ^ { \circ }\) with the downward vertical and \(B\) moves with constant angular speed \(\sqrt { } ( 2 g k )\), where \(k\) is a constant. The tension in the string is \(3 m g\). By modelling \(B\) as a particle. find

1. the value of \(\alpha\),
2. the length of the string.

3. A particle \(P\) of mass 2.5 kg moves along the positive \(x\)-axis. It moves away from a fixed origin \(O\), under the action of a force directed away from \(O\). When \(O P = x\) metres the magnitude of the force is \(2 \mathrm { e } ^ { - 0.1 x }\) newtons and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(x = 0 , v = 2\). Find

1. \(v ^ { 2 }\) in terms of \(x\),
2. the value of \(x\) when \(v = 4\).
3. Give a reason why the speed of \(P\) does not exceed \(\sqrt { } 20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4. A light elastic string \(A B\) of natural length 1.5 m has modulus of elasticity 20 N . The end \(A\) is fixed to a point on a smooth horizontal table. A small ball \(S\) of mass 0.2 kg is attached to the end \(B\). Initially \(S\) is at rest on the table with \(A B = 1.5 \mathrm {~m}\). The ball \(S\) is then projected horizontally directly away from \(A\) with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By modelling \(S\) as a particle,

1. find the speed of \(S\) when \(A S = 2 \mathrm {~m}\). When the speed of \(S\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the string breaks.
2. Find the tension in the string immediately before the string breaks.
   (5)

5.
\includegraphics[max width=\textwidth, alt={}, center]{6d6b6de9-ec06-4b55-8dc6-4923a3554ffa-4_1185_1059_239_416} A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre \(O\) of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is \(r\) and the radius of the base of the cone is \(2 r\). The height of the cone and the height of the cylinder are each \(h\). The centre of mass of the model is at the point \(G\).

1. Show that \(O G = \frac { 1 } { 14 } h\).
   (8) The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle \(\alpha\) with the horizontal, where tan \(\alpha = \frac { 7 } { 26 }\). The desk top is rough enough to prevent slipping and the model is about to topple.
2. Find \(r\) in terms of \(h\).
   (4)

6. A light elastic string, of natural length \(4 a\) and modulus of elasticity \(8 m g\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\).

1. Find the distance \(A O\).
   (2) The particle is now pulled down to a point \(C\) vertically below \(O\), where \(O C = d\). It is released from rest. In the subsequent motion the string does not become slack.
2. Show that \(P\) moves with simple harmonic motion of period \(\pi \sqrt { \left( \frac { 2 a } { g } \right) }\). The greatest speed of \(P\) during this motion is \(\frac { 1 } { 2 } \sqrt { } ( g a )\).
3. Find \(d\) in terms of \(a\).
   (3) Instead of being pulled down a distance \(d\), the particle is pulled down a distance \(a\). Without further calculation,
4. describe briefly the subsequent motion of \(P\).
   (2)

7. \begin{figure}[h] \end{figure} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is hanging at the point \(A\), which is vertically below \(O\). It is projected horizontally with speed \(u\). When the particle is at the point \(P , \angle A O P = \theta\), as shown in Fig. 3. The string oscillates through an angle \(\alpha\) on either side of \(O A\) where \(\cos \alpha = \frac { 2 } { 3 }\).

1. Find \(u\) in terms of g and \(l\). When \(\angle A O P = \theta\), the tension in the string is \(T\).
2. Show that \(T = \frac { m g } { 3 } ( 9 \cos \theta - 4 )\).
3. Find the range of values of \(T\). END