Edexcel M3 (Mechanics 3) 2022 January

1. A light elastic string \(A B\) has natural length \(11 a\) and modulus of elasticity \(6 m g\)

A particle of mass \(4 m\) is attached to the point \(C\) on the string where \(A C = 8 a\) and a particle of mass \(2 m\) is attached to the end \(B\) \begin{figure}[h] \end{figure} The end \(A\) of the string is attached to a fixed point and the string hangs vertically below \(A\) with the particle of mass \(4 m\) in equilibrium at the point \(P\) and the particle of mass \(2 m\) in equilibrium at the point \(Q\), as shown in Figure 1.

1. Find the length \(A P\)
2. Find the length \(P Q\)

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(A\) which lies above a smooth horizontal table. The particle \(P\) moves in a horizontal circle on the table with the string taut. The centre of the circle is the point \(O\) on the table, where \(A O\) is vertical and the string makes a constant angle \(\theta ^ { \circ }\) with \(A O\), as shown in Figure 2. Given that \(P\) moves with constant angular speed \(\sqrt { \frac { 2 g } { a } }\), find the range of possible values of \(\theta\)

1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, where \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\)

The acceleration of \(P\) has magnitude \(\frac { 2 } { ( 2 x + 1 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) and is directed towards \(O\)
When \(t = 0 , P\) passes through \(O\) in the positive \(x\) direction with speed \(1 \mathrm {~ms} ^ { - 1 }\)

1. Find \(v\) in terms of \(x\)
2. Show that \(x = \frac { 1 } { 2 } ( \sqrt { ( 4 t + 1 ) } - 1 )\)

1. A uniform solid hemisphere \(H\) has radius \(r\) and centre \(O\)
   1. Show that the centre of mass of \(H\) is \(\frac { 3 r } { 8 }\) from \(O\)
   $$\left[ \text { You may assume that the volume of } H \text { is } \frac { 2 \pi r ^ { 3 } } { 3 } \right]$$ A uniform solid \(S\), shown below in Figure 3, is formed by attaching a uniform solid right circular cylinder of height \(h\) and radius \(r\) to \(H\), so that one end of the cylinder coincides with the plane face of \(H\). The point \(A\) is the point on \(H\) such that \(O A = r\) and \(O A\) is perpendicular to the plane face of \(H\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-12_592_791_909_660} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}
2. Show that the distance of the centre of mass of \(S\) from \(A\) is $$\frac { 5 r ^ { 2 } + 12 r h + 6 h ^ { 2 } } { 8 r + 12 h }$$ The solid \(S\) can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.
3. Find \(r\) in terms of \(h\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at rest vertically below \(O\), with the string taut, as shown in Figure 4. The particle is then projected horizontally with speed \(u\), where \(u > \sqrt { 2 a g }\)
Air resistance is modelled as being negligible.
At the instant when the string makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\) and the string goes slack.

1. Show that \(3 v ^ { 2 } = u ^ { 2 } - 2 a g\) From the instant when the string goes slack to the instant when \(O P\) is next horizontal, \(P\) moves as a projectile. The time from the instant when the string goes slack to the instant when \(O P\) is next horizontal is \(T\) Given that \(\theta = 30 ^ { \circ }\)
2. show that \(T = \frac { 2 v } { g }\)
3. Hence, show that the string goes taut again when it is next horizontal.

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length 2l. The other end of the spring is attached to a fixed point \(A\). The particle \(P\) hangs in equilibrium vertically below \(A\), at the point \(E\) where \(A E = 6 l\). The particle \(P\) is then raised a vertical distance \(2 l\) and released from rest.

Air resistance is modelled as being negligible.

1. Show that \(P\) moves with simple harmonic motion of period \(T\) where $$T = 4 \pi \sqrt { \frac { l } { g } }$$
2. Find, in terms of \(m , l\) and \(g\), the kinetic energy of \(P\) as it passes through \(E\)
3. Find, in terms of \(T\), the exact time from the instant when \(P\) is released to the instant when \(P\) has moved a distance 31 .

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(O\) on a rough plane which is inclined to the horizontal at an angle \(\alpha\) The string lies along a line of greatest slope of the plane.
The particle \(P\) is held at rest on the plane at the point \(A\), where \(O A = a\), as shown in Figure 5. The particle \(P\) is released from \(A\) and slides down the plane, coming to rest at the point \(B\). The coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \tan \alpha\) Air resistance is modelled as being negligible.

1. Show that \(A B = a ( \sin \alpha - \mu \cos \alpha )\). Given that \(\tan \alpha = \frac { 3 } { 4 }\) and \(\mu = \frac { 1 } { 2 }\)
2. find, in terms of \(a\) and \(g\), the maximum speed of \(P\) as it moves from \(A\) to \(B\)
3. Describe the motion of \(P\) after it reaches \(B\), justifying your answer.