Edexcel FM2 (Further Mechanics 2) 2019 June

1. \begin{figure}[h] \end{figure} A hemispherical shell of radius \(a\) is fixed with its rim uppermost and horizontal. A small bead, \(B\), is moving with constant angular speed, \(\omega\), in a horizontal circle on the smooth inner surface of the shell. The centre of the path of \(B\) is at a distance \(\frac { 1 } { 4 } a\) vertically below the level of the rim of the hemisphere, as shown in Figure 1. Find the magnitude of \(\omega\), giving your answer in terms of \(a\) and \(g\).

1. A particle, \(P\), of mass 0.4 kg is moving along the positive \(x\)-axis, in the positive \(x\) direction under the action of a single force. At time \(t\) seconds, \(t > 0 , P\) is \(x\) metres from the origin \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force is acting in the direction of \(x\) increasing and has magnitude \(\frac { k } { v }\) newtons, where \(k\) is a constant.

At \(x = 3 , v = 2\) and at \(x = 6 , v = 2.5\)

1. Show that \(v ^ { 3 } = \frac { 61 x + 9 } { 24 }\) The time taken for the speed of \(P\) to increase from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
2. Use algebraic integration to show that \(T = \frac { 81 } { 61 }\)

1. Numerical (calculator) integration is not acceptable in this question.

\begin{figure}[h] \end{figure} The shaded region \(O A B\) in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 4\) and the curve with equation \(y = \frac { 1 } { 4 } ( x - 2 ) ^ { 3 } + 2\). The point \(A\) has coordinates (4, 4) and the point \(B\) has coordinates \(( 4,0 )\). A uniform lamina \(L\) has the shape of \(O A B\). The unit of length on both axes is one centimetre. The centre of mass of \(L\) is at the point with coordinates \(( \bar { x } , \bar { y } )\). Given that the area of \(L\) is \(8 \mathrm {~cm} ^ { 2 }\),

1. show that \(\bar { y } = \frac { 8 } { 7 }\) The lamina is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\).

1. A flagpole, \(A B\), is 4 m long. The flagpole is modelled as a non-uniform rod so that, at a distance \(x\) metres from \(A\), the mass per unit length of the flagpole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\), is given by \(m = 18 - 3 x\).
   1. Show that the mass of the flagpole is 48 kg .
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-12_515_439_502_806} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The end \(A\) of the flagpole is fixed to a point on a vertical wall. A cable has one end attached to the midpoint of the flagpole and the other end attached to a point on the wall that is vertically above \(A\). The cable is perpendicular to the flagpole. The flagpole and the cable lie in the same vertical plane that is perpendicular to the wall. A small ball of mass 4 kg is attached to the flagpole at \(B\). The cable holds the flagpole and ball in equilibrium, with the flagpole at \(45 ^ { \circ }\) to the wall, as shown in Figure 3. The tension in the cable is \(T\) newtons.
   The cable is modelled as a light inextensible string and the ball is modelled as a particle.
2. Using the model, find the value of \(T\).
3. Give a reason why the answer to part (b) is not likely to be the true value of \(T\).

5. \begin{figure}[h] \end{figure} The region \(R\), shown shaded in Figure 4, is bounded by part of the curve with equation \(y ^ { 2 } = 2 x\), the line with equation \(y = 2\) and the \(y\)-axis. The unit of length on both axes is one centimetre. A uniform solid, \(S\), is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(y\)-axis.
Given that the volume of \(S\) is \(\frac { 8 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),

1. show that the centre of mass of \(S\) is \(\frac { 1 } { 3 } \mathrm {~cm}\) from its plane face. A uniform solid cylinder, \(C\), has base radius 2 cm and height 4 cm . The cylinder \(C\) is attached to \(S\) so that the plane face of \(S\) coincides with a plane face of \(C\), to form the paperweight \(P\), shown in Figure 5. The density of the material used to make \(S\) is three times the density of the material used to make \(C\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-16_572_456_1617_758} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The plane face of \(P\) rests in equilibrium on a desk lid that is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The lid is sufficiently rough to prevent \(P\) from slipping. Given that \(P\) is on the point of toppling,
2. find the value of \(\theta\).

1. The points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 4.5 \mathrm {~m}\).

A light elastic string has natural length 1.5 m and modulus of elasticity 15 N . One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). A particle, \(P\), of mass 0.2 kg , is attached to the stretched string so that \(A P B\) is a straight line and \(A P = 1.5 \mathrm {~m}\). The particle rests in equilibrium on the surface. The particle is now moved directly towards \(A\) and is held on the surface so \(A P B\) is a straight line with \(A P = 1 \mathrm {~m}\). The particle is released from rest.

1. Prove that \(P\) moves with simple harmonic motion.
2. Find
   1. the maximum speed of \(P\) during the motion,
   2. the maximum acceleration of \(P\) during the motion.
3. Find the total time, in each complete oscillation of \(P\), for which the speed of \(P\) is greater than \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. A particle, \(P\), of mass \(m\) is attached to one end of a light rod of length \(L\). The other end of the rod is attached to a fixed point \(O\) so that the rod is free to rotate in a vertical plane about \(O\). The particle is held with the rod horizontal and is then projected vertically downwards with speed \(u\). The particle first comes to instantaneous rest at the point \(A\).
   1. Explain why the acceleration of \(P\) at \(A\) is perpendicular to \(O A\).
   At the instant when \(P\) is at the point \(A\) the acceleration of \(P\) is in a direction making an angle \(\theta\) with the horizontal. Given that \(u ^ { 2 } = \frac { 2 g L } { 3 }\),
2. find
   1. the magnitude of the acceleration of \(P\) at the point \(A\),
   2. the size of \(\theta\).
3. Find, in terms of \(m\) and \(g\), the magnitude of the tension in the rod at the instant when \(P\) is at its lowest point.