Edexcel FM2 (Further Mechanics 2) 2020 June

1. Three particles of masses \(3 m\), \(4 m\) and \(2 m\) are placed at the points \(( - 2,2 ) , ( 3,1 )\) and ( \(p , p\) ) respectively.

The value of \(p\) is such that the distance of the centre of mass of the three particles from the point ( 0,0 ) is as small as possible. Find the value of \(p\).

2. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A uniform plane figure \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = \ln 5\), the curve with equation \(y = 8 \mathrm { e } ^ { - x }\) and the line with equation \(x = \ln 2\). The unit of length on each axis is one metre. The area of \(R\) is \(2.4 \mathrm {~m} ^ { 2 }\)
The centre of mass of \(R\) is at the point with coordinates \(( \bar { x } , \bar { y } )\).

1. Use algebraic integration to show that \(\bar { y } = 1.4\) Figure 2 shows a uniform lamina \(A B C D\), which is the same size and shape as \(R\). The lamina is freely suspended from \(C\) and hangs in equilibrium with \(C B\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\)

1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the direction of \(x\) increasing. At time \(t\) seconds \(( t \geqslant 0 ) , P\) is \(x\) metres from the origin \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant force acting on \(P\) is directed towards \(O\) and has magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a positive constant.

When \(x = 1 , v = 4\) and when \(x = 2 , v = 2\)

1. Show that \(v = a b ^ { x }\), where \(a\) and \(b\) are constants to be found. The time taken for the speed of \(P\) to decrease from \(4 \mathrm {~ms} ^ { - 1 }\) to \(2 \mathrm {~ms} ^ { - 1 }\) is \(T\) seconds.
2. Show that \(T = \frac { 1 } { 4 \ln 2 }\)

4. \begin{figure}[h] \end{figure} A uniform solid cylinder of base radius \(r\) and height \(\frac { 4 } { 3 } r\) has the same density as a uniform solid hemisphere of radius \(r\). The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid \(S\) shown in Figure 3. The point \(O\) is the centre of the plane face of \(S\).

1. Show that the distance from \(O\) to the centre of mass of \(S\) is \(\frac { 73 } { 72 } r\) The solid \(S\) is placed with its plane face on a rough horizontal plane. The coefficient of friction between \(S\) and the plane is \(\mu\). A horizontal force \(P\) is applied to the highest point of \(S\). The magnitude of \(P\) is gradually increased.
2. Find the range of values of \(\mu\) for which \(S\) will slide before it starts to tilt.

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm . The other end of the string is attached to a fixed point \(A\) that is vertically above the point \(O\) on a smooth horizontal table, such that \(O A = 40 \mathrm {~cm}\). The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre \(O\), as shown in Figure 4. The particle is moving with a constant angular speed of 3 radians per second.

1. Find (i) the tension in the string,
   (ii) the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
2. Find the angular speed of \(P\) at the instant \(P\) loses contact with the table.

6. \begin{figure}[h] \end{figure} A particle \(P\) 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 held with the string taut and \(O P\) horizontal. The particle is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 9 } { 5 } \mathrm { gl }\). When \(O P\) has turned through an angle \(\alpha\) and the string is still taut, the speed of \(P\) is \(v\), as shown in Figure 5. At this instant the tension in the string is \(T\).

1. Show that \(T = 3 m g \sin \alpha + \frac { 9 } { 5 } m g\)
2. Find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the string goes slack.
3. Find, in terms of \(l\), the greatest vertical height reached by \(P\) above the level of \(O\).

1. A light elastic spring has natural length \(l\) and modulus of elasticity \(4 m g\). A particle \(P\) of mass \(m\) is attached to one end of the spring. The other end of the spring is attached to a fixed point \(A\). The point \(B\) is vertically below \(A\) with \(A B = \frac { 7 } { 4 } l\). The particle \(P\) is released from rest at \(B\).
   1. Show that \(P\) moves with simple harmonic motion with period \(\pi \sqrt { \frac { l } { g } }\)
   2. Find, in terms of \(m , l\) and \(g\), the maximum kinetic energy of \(P\) during the motion.
   3. Find the time within each complete oscillation for which the length of the spring is less than \(l\).