Edexcel M3 (Mechanics 3) 2021 January

1. \begin{figure}[h] \end{figure} The region \(R\), shown shaded in Figure 1, is bounded by the curve with equation \(y = \frac { 1 } { x }\), the line with equation \(x = 1\), the positive \(x\)-axis and the line with equation \(x = a\) where \(a > 1\) A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the volume of \(S\) is $$\pi \left( 1 - \frac { 1 } { a } \right)$$
2. Find the \(x\) coordinate of the centre of mass of \(S\).

2. A particle \(P\) of mass \(m\) is at a distance \(x\) above the surface of the Earth. The Earth exerts a gravitational force on \(P\). This force is directed towards the centre of the Earth. The magnitude of this force is inversely proportional to the square of the distance of \(P\) from the centre of the Earth. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).

1. Show that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { ( x + R ) ^ { 2 } }\) A particle is released from rest from a point above the surface of the Earth. When the particle is at a distance \(R\) above the surface of the Earth, the particle has speed \(U\). Air resistance is modelled as being negligible.
2. Find, in terms of \(U , g\) and \(R\), the speed of the particle when it strikes the surface of the Earth.
   

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3. \begin{figure}[h] \end{figure} A fairground ride consists of a cabin \(C\) that travels in a horizontal circle with a constant angular speed about a fixed vertical central axis. The cabin is attached to one end of each of two rigid arms, each of length 5 m . The other end of the top arm is attached to the fixed point \(A\) at the top of the central axis of the ride. The other end of the lower arm is attached to the fixed point \(B\) on the central axis, where \(A B\) is 8 m , as shown in Figure 2. Both arms are free to rotate about the central axis. The arms are modelled as light inextensible rods. The cabin, together with the people inside, is modelled as a particle. The cabin completes one revolution every 2 seconds. Given that the combined mass of the cabin and the people is 600 kg ,

1. find
   1. the tension in the upper arm of the ride,
   2. the tension in the lower arm of the ride. In a refined model, it is assumed that both arms stretch to a length of 5.1 m .
2. State how this would affect the sum of the tensions in the two arms, justifying your answer.

4. \begin{figure}[h] \end{figure} A uniform right solid cone \(C\) has diameter \(6 a\) and height \(8 a\), as shown in Figure 3.
The solid \(S\) is formed by removing a cone of height \(4 a\) from the top of \(C\) and then removing an identical, inverted cone. The vertex of the removed cone is at the point \(O\) in the centre of the base of \(C\), as shown in Figure 4. \begin{figure}[h] \end{figure}

1. Find the distance of the centre of mass of \(S\) from \(O\).
   (5) The point \(A\) lies on the circumference of the base of \(S\) and the point \(B\) lies on the circumference of the top of \(S\). The points \(O\), \(A\) and \(B\) all lie in the same vertical plane, as shown in Figure 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_248_449_1845_749} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The solid \(S\) is freely suspended from the point \(B\) and hangs in equilibrium.
2. Find the size of the angle that \(A B\) makes with the downward vertical.

5. \begin{figure}[h] \end{figure} The fixed points, \(A\) and \(B\), are a distance \(10 a\) apart, with \(B\) vertically above \(A\). One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(2 m g\), is attached to a particle \(P\) of mass \(m\) and the other end is attached to \(A\). One end of another light elastic string, of natural length \(4 a\) and modulus of elasticity \(6 m g\), is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(C\), as shown in Figure 6.

1. Show that each string has an extension of \(2 a\).
   (5) The particle \(P\) is now pulled down vertically, so that it is a distance \(a\) below \(C\) and then released from rest.
2. Show that in the subsequent motion, \(P\) performs simple harmonic motion.
3. Find, in terms of \(a\) and \(g\), the speed of \(P\) when it is a distance \(\frac { 7 } { 2 } a\) above \(A\).

6. \begin{figure}[h] \end{figure} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is attached to a fixed point \(O\). The particle is initially held at the point \(A\) with the string taut and \(O A\) making an angle of \(60 ^ { \circ }\) with the downward vertical. The particle is then projected upwards with a speed of \(3 \sqrt { a g }\), perpendicular to \(O A\), in the vertical plane containing \(O A\), as shown in Figure 7. In an initial model of the motion of the particle, it is assumed that the string does not break. Using this model,

1. show that the particle performs complete vertical circles. In a refined model it is assumed that the string will break if the tension in it exceeds 7 mg . Using this refined model,
2. show that the particle still performs complete vertical circles.
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7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 3 N . The other end of the string is attached to a fixed point \(O\) on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 2 } { 7 }\) The coefficient of friction between \(P\) and the plane is \(\frac { \sqrt { 5 } } { 5 }\) The particle \(P\) is initially at rest at the point \(O\), as shown in Figure 8. The particle \(P\) then receives an impulse of magnitude 4 Ns, directed up a line of greatest slope of the plane. The particle \(P\) moves up the plane and comes to rest at the point \(A\).

1. Find the extension of the elastic string when \(P\) is at \(A\).
2. Show that the particle does not remain at rest at \(A\).