Edexcel M3 (Mechanics 3) 2019 January

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement of \(P\) from the origin \(O\) is \(x\) metres and the acceleration of \(P\) is \(\left( \frac { 7 } { 2 } - 2 x \right) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the positive \(x\) direction. At time \(t = 0 , P\) passes through \(O\) moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) in the positive \(x\) direction. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
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2. \begin{figure}[h] \end{figure} A small ball \(P\) of mass \(m\) is attached to the midpoint of a light inextensible string of length \(2 a\). The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = a\), as shown in Figure 1. The system rotates about the line \(A B\) with constant angular speed \(\omega\). The ball moves in a horizontal circle with both parts of the string taut. The tension in the string must be less than \(3 m g\) otherwise the string will break. Given that the time taken by the ball to complete one revolution is \(S\), show that $$\pi \sqrt { \frac { a } { g } } < S < \pi \sqrt { \frac { k a } { g } }$$ stating the value of the constant \(k\).

3. A particle \(P\) is moving in a straight line with simple harmonic motion between two points \(A\) and \(B\), where \(A B\) is \(2 a\) metres. The point \(C\) lies on the line \(A B\) and \(A C = \frac { 1 } { 2 } a\) metres. The particle passes through \(C\) with speed \(\frac { 3 a \sqrt { 3 } } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the period of the motion. The maximum magnitude of the acceleration of \(P\) is \(45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find
2. the value of \(a\),
3. the maximum speed of \(P\). The point \(D\) lies on \(A B\) and \(P\) takes a quarter of one period to travel directly from \(C\) to \(D\).
4. Find the distance CD.

4. \begin{figure}[h] \end{figure} The ends of a light elastic string, of natural length \(4 l\) and modulus of elasticity \(\lambda\), are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 4 l\). A particle \(P\) of mass \(2 m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at a distance \(\frac { 3 } { 2 } l\) vertically below the midpoint of \(A B\), as shown in Figure 2.

1. Show that \(\lambda = \frac { 20 } { 3 } m g\). The particle is pulled vertically downwards from its equilibrium position until the total length of the string is 6l. The particle is then released from rest.
2. Show that \(P\) comes to instantaneous rest before reaching the line \(A B\).

5. \begin{figure}[h] \end{figure} The region \(R\), shown shaded in Figure 3, is bounded by the circle with centre \(O\) and radius \(r\), the line with equation \(x = \frac { 3 } { 5 } r\) and the \(x\)-axis. The region is rotated through one complete revolution about the \(x\)-axis to form a uniform solid \(S\).

1. Use algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 48 } { 65 } r\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_394_643_1311_653} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} A bowl is made from a uniform solid hemisphere of radius 6 cm by removing a hemisphere of radius 5 cm . Both hemispheres have the same centre \(A\) and the same axis of symmetry. The bowl is fixed with its open plane face uppermost and horizontal. Liquid is poured into the bowl. The depth of the liquid is 2 cm , as shown in Figure 4. The mass of the empty bowl is \(5 M \mathrm {~kg}\) and the mass of the liquid is \(2 M \mathrm {~kg}\).
2. Find, to 3 significant figures, the distance from \(A\) to the centre of mass of the bowl with its liquid.

6. \begin{figure}[h] \end{figure} Figure 5 shows a hollow sphere, with centre \(O\) and internal radius \(a\), which is fixed to a horizontal surface. A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { \frac { 7 a g } { 2 } }\) from the lowest point \(A\) of the inner surface of the sphere. The particle moves in a vertical circle with centre \(O\) on the smooth inner surface of the sphere. The particle passes through the point \(B\), on the inner surface of the sphere, where \(O B\) is horizontal.

1. Find, in terms of \(m\) and \(g\), the normal reaction exerted on \(P\) by the surface of the sphere when \(P\) is at \(B\). The particle leaves the inner surface of the sphere at the point \(C\), where \(O C\) makes an angle \(\theta , \theta > 0\), with the upward vertical.
2. Show that, after leaving the surface of the sphere at \(C\), the particle is next in contact with the surface at \(A\).

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