Edexcel FM2 (Further Mechanics 2) 2023 June

1. Three particles of masses \(3 m , 4 m\) and \(k m\) are positioned at the points with coordinates ( \(2 a , 3 a\) ), ( \(a , 5 a\) ) and ( \(2 \mu a , \mu a\) ) respectively, where \(k\) and \(\mu\) are constants.

The centre of mass of the three particles is at the point with coordinates \(( 2 a , 4 a )\).
Find (i) the value of \(k\)
(ii) the value of \(\mu\)

1. A particle of mass 2 kg is moving in a straight line on a smooth horizontal surface under the action of a horizontal force of magnitude \(F\) newtons.

At time \(t\) seconds \(( t > 0 )\),

• the particle is moving with speed \(v \mathrm {~ms} ^ { - 1 }\)
• \(F = 2 + v\)

The time taken for the speed of the particle to increase from \(5 \mathrm {~ms} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.

1. Show that \(T = 2 \ln \frac { 12 } { 7 }\) The distance moved by the particle as its speed increases from \(5 \mathrm {~ms} ^ { - 1 }\) to \(10 \mathrm {~ms} ^ { - 1 }\) is \(D\) metres.
2. Find the exact value of \(D\).

1. \hspace{0pt} [In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]

\begin{figure}[h] \end{figure} Five pieces of a uniform wire are joined together to form the rigid framework \(O A B C O\) shown in Figure 1, where

• \(O A , O B\) and \(B C\) are straight, with \(O A = O B = B C = r\)
• arc \(A B\) is one quarter of a circle with centre \(O\) and radius \(r\)
• arc \(O C\) is one quarter of a circle of radius \(r\)
• all five pieces of wire lie in the same plane
  1. Show that the centre of mass of arc \(A B\) is a distance \(\frac { 2 r } { \pi }\) from \(O A\).

Given that the distance of the centre of mass of the framework from \(O A\) is \(d\),

show that \(\mathrm { d } = \frac { 7 r } { 2 ( 3 + ) }\) The framework is freely pivoted at \(A\).
The framework is held in equilibrium, with \(A O\) vertical, by a horizontal force of magnitude \(F\) which is applied to the framework at \(C\). Given that the weight of the framework is \(W\)

find \(F\) in terms of \(W\)

4. \begin{figure}[h] \end{figure} A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\). A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,

• the speed of \(P\) is \(v\)
• the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
  1. Show that \(\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )\)
  2. Find, in terms of \(a\) and \(g\), the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

5. \begin{figure}[h] \end{figure} A uniform lamina \(O A B\) is modelled by the finite region bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 9 - x ^ { 2 }\), for \(x \geqslant 0\), as shown shaded in Figure 3. The unit of length on both axes is 1 m . The area of the lamina is \(18 \mathrm {~m} ^ { 2 }\)

1. Show that the centre of mass of the lamina is 3.6 m from \(\boldsymbol { O B }\).
   [0pt] [ Solutions relying on calculator technology are not acceptable.] A light string has one end attached to the lamina at \(O\) and the other end attached to the ceiling. A second light string has one end attached to the lamina at \(A\) and the other end attached to the ceiling.
   The lamina hangs in equilibrium with the strings vertical and \(O A\) horizontal.
   The weight of the lamina is \(W\)
   The tension in the string attached to the lamina at \(A\) is \(\lambda W\)
2. Find the value of \(\lambda\)

6. \begin{figure}[h] \end{figure} A hollow right circular cone, of internal base radius 0.6 m and height 0.8 m , is fixed with its axis vertical and its vertex \(V\) pointing downwards, as shown in Figure 4. A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle of radius 0.5 m on the rough inner surface of the cone. The particle \(P\) moves with constant angular speed \(\omega\) rads \(^ { - 1 }\)
The coefficient of friction between the particle \(P\) and the inner surface of the cone is 0.25 Find the greatest possible value of \(\omega\)

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The shaded region shown in Figure 5 is bounded by the line with equation \(x = a\) and the curve with equation \(x ^ { 2 } + y ^ { 2 } = 4 a ^ { 2 }\) This shaded region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid is used to model a dome with height \(a\) metres and base radius \(\sqrt { 3 } a\) metres.
The dome is modelled as being non-uniform with the mass per unit volume of the dome at the point \(( x , y , z )\) equal to \(\frac { \lambda } { x ^ { 2 } } \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), where \(a \leqslant x \leqslant 2 a\) and \(\lambda\) is a constant.

1. Show that the distance of the centre of mass of the dome from the centre of its plane face is \(\left( 4 \ln 2 - \frac { 5 } { 2 } \right) a\) metres. A solid uniform right circular cone has base radius \(\sqrt { 3 } a\) metres and perpendicular height \(4 a\) metres. A toy is formed by attaching the plane surface of the dome to the plane surface of the cone, as shown in Figure 6. The weight of the cone is \(k W\) and the weight of the dome is \(2 W\)
   The centre of mass of the toy is a distance \(d\) metres from the plane face of the dome.
2. Show that \(d = \frac { | k + 5 - 8 \ln 2 | } { 2 + k } a\) The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle \(\alpha\) to the downward vertical.
   Given that \(\tan \alpha = \frac { 1 } { 2 \sqrt { 3 } }\)
3. find the exact value of \(k\).

8. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 6 \mathrm {~m}\).
A particle \(P\) has mass 0.3 kg .
One end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N , is attached to \(P\), and the other end is attached to \(A\). One end of another light elastic string, of natural length 2 m and modulus of elasticity 40 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) is at rest in equilibrium at the point \(E\) on the surface, as shown in Figure 7.

1. Show that \(E B = \frac { 8 } { 3 } \mathrm {~m}\). The particle \(P\) is now held at the midpoint of \(A B\) and released from rest.
2. Show that \(P\) oscillates with simple harmonic motion about the point \(E\). The time between the instant when \(P\) is released and the instant when it first returns to the point \(E\) is \(S\) seconds.
3. Find the exact value of \(S\).
4. Find the length of time during one oscillation for which the speed of \(P\) is more than \(2 \mathrm {~ms} ^ { - 1 }\)