Edexcel FM2 (Further Mechanics 2) 2024 June

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A particle \(P\) moves along a straight line. Initially \(P\) is at rest at the point \(O\) on the line. At time \(t\) seconds, where \(t \geqslant 0\)

• the displacement of \(P\) from \(O\) is \(x\) metres
• the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction
• the acceleration of \(P\) is \(\frac { 96 } { ( 3 t + 5 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction
  1. Show that, at time \(t\) seconds, \(v = p - \frac { q } { ( 3 t + 5 ) ^ { 2 } }\), where \(p\) and \(q\) are constants to be determined.
  2. Find the limiting value of \(v\) as \(t\) increases.
  3. Find the value of \(x\) when \(t = 2\)

2. \begin{figure}[h] \end{figure} A uniform rod of length \(28 a\) is cut into seven identical rods each of length \(4 a\). These rods are joined together to form the rigid framework \(A B C D E A\) shown in Figure 1. All seven rods lie in the same plane.
The distance of the centre of mass of the framework from \(E D\) is \(d\).

1. Show that \(d = \frac { 8 \sqrt { 3 } } { 7 } a\) The weight of the framework is \(W\).
   The framework is freely pivoted about a horizontal axis through \(C\).
   The framework is held in equilibrium in a vertical plane, with \(A C\) vertical and \(A\) below \(C\), by a horizontal force that is applied to the framework at \(A\). The force acts in the same vertical plane as the framework and has magnitude \(F\).
2. Find \(F\) in terms of \(W\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a hemispherical bowl of internal radius \(10 d\) that is fixed with its circular rim horizontal. The centre of the circular rim is at the point \(O\).
A particle \(P\) moves with constant angular speed on the smooth inner surface of the bowl. The particle \(P\) moves in a horizontal circle with radius \(8 d\) and centre \(C\).

1. Find, in terms of \(g\), the exact magnitude of the acceleration of \(P\). The time for \(P\) to complete one revolution is \(T\).
2. Find \(T\) in terms of \(d\) and \(g\).

4. \begin{figure}[h] \end{figure} A uniform lamina \(O A B\) is in the shape of the region \(R\).
Region \(R\) lies in the first quadrant and is bounded by the curve with equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 36 } = 1\), the \(x\)-axis, and the \(y\)-axis, as shown shaded in Figure 3. The point \(A\) is the point of intersection of the curve and the \(x\)-axis.
The point \(B\) is the point of intersection of the curve and the \(y\)-axis.
One unit on each axis represents 1 m .
The area of \(R\) is \(6 \pi\)
The centre of mass of \(R\) lies at the point with coordinates \(( \bar { x } , \bar { y } )\)

1. Use algebraic integration to show that \(\bar { x } = \frac { 16 } { 3 \pi }\)
2. Use algebraic integration to find the exact value of \(\bar { y }\) The lamina is freely suspended from \(A\) and hangs in equilibrium with \(O A\) at angle \(\theta ^ { \circ }\) to the downward vertical.
3. Find the value of \(\theta\)

1. A particle \(P\) moves in a straight line with simple harmonic motion about a fixed point \(O\). The magnitude of the greatest acceleration of \(P\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

When \(P\) is 0.3 m from \(O\), the speed of \(P\) is \(2.4 \mathrm {~ms} ^ { - 1 }\)
The amplitude of the motion is \(a\) metres.

1. Show that \(a = 0.5\)
2. Find the greatest speed of \(P\). During one oscillation, the speed of \(P\) is at least \(2 \mathrm {~ms} ^ { - 1 }\) for \(S\) seconds.
3. Find the value of \(S\).

6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The shaded region, shown in Figure 4, is bounded by the \(x\)-axis, the line with equation \(x = 6\), the line with equation \(y = 2\) and the \(y\)-axis. This region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { x }\)-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm . The mass per unit volume of the cylinder at the point \(( x , y , z )\) is \(\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }\), where \(0 \leqslant x \leqslant 6\) and \(\lambda\) is a constant.

1. Show that the mass of the cylinder is \(120 \lambda \pi \mathrm {~kg}\).
2. Show that the centre of mass of the cylinder is 3.6 cm from \(O\). The point \(O\) is the centre of one end of the cylinder. The point \(A\) is the centre of the other end of the cylinder. A uniform solid hemisphere of radius 3 cm has density \(\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }\). The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point \(A\) on the cylinder to form the model shown in Figure 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The model is placed with the end containing \(O\) on a rough inclined plane which is inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
3. Find the value of \(\alpha\).

7. \begin{figure}[h] \end{figure} A smooth solid hemisphere has radius \(r\) and the centre of its plane face is \(O\).
The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6.
A small stone is at the point \(A\), the highest point on the surface of the hemisphere. The stone is projected horizontally from \(A\) with speed \(U\).
The stone is still in contact with the hemisphere at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical.
The speed of the stone at the instant it reaches \(B\) is \(v\).
The stone is modelled as a particle \(P\) and air resistance is modelled as being negligible.

1. Use the model to find \(v ^ { 2 }\) in terms of \(U , r , g\) and \(\theta\) When \(P\) leaves the surface of the hemisphere, the speed of \(P\) is \(W\).
   Given that \(U = \sqrt { \frac { 2 r g } { 3 } }\)
2. show that \(W ^ { 2 } = \frac { 8 } { 9 } r g\) After leaving the surface of the hemisphere, \(P\) moves freely under gravity until it hits the ground.
3. Find the speed of \(P\) as it hits the ground, giving your answer in terms of \(r\) and \(g\). At the instant when \(P\) hits the ground it is travelling at \(\alpha ^ { \circ }\) to the horizontal.
4. Find the value of \(\alpha\).