Edexcel M3 (Mechanics 3) 2022 January

1. \begin{figure}[h] \end{figure} A uniform lamina is in the shape of the region \(R\).
Region \(R\) is bounded by the curve with equation \(y = x ( x + a )\) where \(a\) is a positive constant, the positive \(x\)-axis and the line with equation \(x = a\), as shown shaded in Figure 1. Find the \(\boldsymbol { y }\) coordinate of the centre of mass of the lamina.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 l\). The other end of the string is attached to a fixed point \(A\) above a smooth horizontal floor. The particle moves in a horizontal circle on the floor with the string taut. The centre \(O\) of the circle is vertically below \(A\) with \(O A = l\), as shown in Figure 2 . The particle moves with constant angular speed \(\omega\) and remains in contact with the floor.
Show that $$\omega \leqslant \sqrt { \frac { g } { l } }$$

1. A particle \(P\) of mass \(m \mathrm {~kg}\) is initially held at rest at the point \(O\) on a smooth inclined plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\)

The particle is released from rest and slides down the plane against a force which acts towards \(O\). The force has magnitude \(\frac { 1 } { 3 } m x ^ { 2 } \mathrm {~N}\), where \(x\) metres is the distance of \(P\) from \(O\).

1. Find the speed of \(P\) when \(x = 2\) The particle first comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\).

4. \begin{figure}[h] \end{figure} A thin uniform right hollow cylinder, of radius \(2 a\) and height \(k a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(2 a\), is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container \(C\). The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of \(C\) is \(O\), as shown in Figure 3.

1. Show that the distance from \(O\) to the centre of mass of \(C\) is $$\frac { \left( k ^ { 2 } + 4 k + 4 \right) } { 2 ( k + 3 ) } a$$ The container is placed with its circular base on a plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(C\) from sliding. The container is on the point of toppling.
2. Find the value of \(k\).

1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds the displacement of \(P\) from the origin \(O\) is \(x\) metres, where \(x = 4 \cos \left( \frac { 1 } { 5 } \pi t \right)\)
   1. Prove that \(P\) is moving with simple harmonic motion.
   2. Find the period of the motion.
   3. State the amplitude of the motion.
   4. Find, in terms of \(\pi\), the maximum speed of \(P\)
   The points \(A\) and \(B\) lie on the \(x\)-axis, on opposite sides of \(O\), with \(O A = 1.5 \mathrm {~m}\) and \(O B = 2.5 \mathrm {~m}\).
2. Find the time taken by \(P\) to move directly from \(A\) to \(B\).

1. A particle \(P\) of mass 1.2 kg is attached to the midpoint of a light elastic string of natural length 0.5 m and modulus of elasticity \(\lambda\) newtons.

The fixed points \(A\) and \(B\) are 0.8 m apart on a horizontal ceiling. One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). Initially \(P\) is held at rest at the midpoint \(M\) of the line \(A B\) and the tension in the string is 30 N .

1. Show that \(\lambda = 50\) The particle is now held at rest at the point \(C\), where \(C\) is 0.3 m vertically below \(M\). The particle is released from rest.
2. Find the magnitude of the initial acceleration of \(P\)
3. Find the speed of \(P\) at the instant immediately before it hits the ceiling.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can rotate freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\). The line \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), where \(\alpha < \frac { \pi } { 2 }\) When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 4.

1. Show that \(v ^ { 2 } = u ^ { 2 } - 2 g l ( \cos \theta - \cos \alpha )\) Given that \(\cos \alpha = \frac { 2 } { 5 }\) and that \(u = \sqrt { 3 g l }\)
2. show that \(P\) moves in a complete vertical circle. As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(k T\)
3. Find the exact value of \(k\)