OCR MEI Further Mechanics Major (Further Mechanics Major) 2023 June

1 A car of mass 800 kg moves in a straight line along a horizontal road.
There is a constant resistance to the motion of the car of magnitude 600 N .
When the car is travelling at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the power developed by the car is 27 kW .
Determine the acceleration of the car when it is travelling at \(15 \mathrm {~ms} ^ { - 1 }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-02_232_609_840_239} Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) and \(B\) is moving away from \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Spheres A and B collide. After this collision A moves with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Determine the possible speeds with which B moves after the collision.

3
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-03_565_757_251_242} The diagram shows a particle P , of mass 0.2 kg , which is attached by a light inextensible string of length 0.75 m to a fixed point O . Particle P moves with constant angular speed \(\omega\) rads \(^ { - 1 }\) in a horizontal circle with centre vertically below O . The string is inclined at an angle \(\theta\) to the vertical. The greatest tension that the string can withstand without breaking is 15 N .

1. Find the greatest possible value of \(\theta\), giving your answer to the nearest degree.
2. Determine the greatest possible value of \(\omega\).

4
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-04_598_696_255_246}
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-04_465_67_294_1023} A rigid lamina of negligible mass is in the form of a rhombus ABCD , where \(\mathrm { AC } = 6 \mathrm {~m}\) and \(\mathrm { BD } = 8 \mathrm {~m}\). Forces of magnitude \(2 \mathrm {~N} , 4 \mathrm {~N} , 3 \mathrm {~N}\) and 5 N act along its sides \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD }\) and DA , respectively, as shown in the diagram. A further force \(\mathbf { F }\) N, acting at A, and a couple of magnitude \(G N m\) are also applied to the lamina so that it is in equilibrium.

1. Determine the magnitude and direction of \(\mathbf { F }\).
2. Determine the value of \(G\).

5 A particle P of mass \(m \mathrm {~kg}\) is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a rough horizontal surface. During the motion of P , a constant frictional force of magnitude \(F \mathrm {~N}\) acts on P . When the velocity of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a force of magnitude \(k v \mathrm {~N}\) due to air resistance, where \(k\) is a constant.

1. Determine the dimensions of \(k\). At time \(T\) s after projection P comes to rest. A formula approximating the value of \(T\) is
   \(\mathrm { T } = \frac { \mathrm { mu } } { \mathrm { F } } - \frac { \mathrm { kmu } ^ { 2 } } { 2 \mathrm {~F} ^ { 2 } } + \frac { 1 } { 3 } \mathrm { k } ^ { 2 } \mathrm {~m} ^ { \alpha } \mathrm { u } ^ { \beta } \mathrm { F } ^ { \gamma }\).
2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).

6 At time \(t\) seconds, where \(t \geqslant 0\), a particle P has position vector \(\mathbf { r }\) metres, where
\(\mathbf { r } = \left( 2 t ^ { 2 } - 12 t + 6 \right) \mathbf { i } + \left( t ^ { 3 } + 3 t ^ { 2 } - 8 t \right) \mathbf { j }\).
The velocity of P at time \(t\) seconds is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\).

1. Find \(\mathbf { v }\) in terms of \(t\).
2. Determine the speed of P at the instant when it is moving parallel to the vector \(\mathbf { i } - 4 \mathbf { j }\).
3. Determine the value of \(t\) when the magnitude of the acceleration of P is \(20.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

7 One end of a rope is attached to a block A of mass 2 kg . The other end of the rope is attached to a second block B of mass 4 kg . Block A is held at rest on a fixed rough ramp inclined at \(30 ^ { \circ }\) to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P , at a distance \(d \mathrm {~m}\) above the ground, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-05_351_851_1274_244} Block A is more than \(d \mathrm {~m}\) from P . The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is \(\frac { 1 } { 2 \sqrt { 3 } }\). The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.

1. Determine, in terms of \(g\) and \(d\), the work done against friction as A moves \(d \mathrm {~m}\) up the ramp.
2. Given that the speed of B immediately before it hits the ground is \(1.75 \mathrm {~ms} ^ { - 1 }\), use the work-energy principle to determine the value of \(d\).
3. Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic.

8
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-06_652_716_258_230} The diagram shows the shaded region R bounded by the curve \(\mathrm { y } = \sqrt { 3 \mathrm { x } + 4 }\), the \(x\)-axis, the \(y\)-axis, and the straight line that passes through the points \(( k , 0 )\) and \(( 4,4 )\), where \(0 < k < 4\). Region R is occupied by a uniform lamina.

1. Determine, in terms of \(k\), an expression for the \(y\)-coordinate of the centre of mass of the lamina. Give your answer in the form \(\frac { \mathrm { a } + \mathrm { bk } } { \mathrm { c } + \mathrm { dk } }\), where \(a , b , c\) and \(d\) are integers to be determined.
2. Show that the \(y\)-coordinate of the centre of mass of the lamina cannot be \(\frac { 3 } { 2 }\).

1. Determine the following, in either order.
   • The components of the velocity of P , parallel and perpendicular to the plane, immediately before P hits the plane at A .
   • The distance OA.
   After P hits the plane at A it continues to move away from O . Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.
2. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree.
   \includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-08_615_759_251_244} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O . The point A lies on the rim of the bowl such that AO makes an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A , with speed \(u\), where \(\mathrm { u } > \sqrt { \frac { \mathrm { gr } } { 2 } }\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A . The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.
3. Determine, in terms of \(m , g , u , r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B . The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4 m g\).
4. Determine, in terms of \(r\), the vertical distance of B above the floor. It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.
5. Show that \(\mathrm { u } ^ { 2 } > 2 \mathrm { gr }\).

11
\includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-09_716_703_251_246} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC , with AB perpendicular to AC , which lies in a vertical plane. The length of AB is 3 cm , and the length of AC is 12 cm . The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle \(\theta\) with the horizontal. A horizontal force of magnitude \(P \mathrm {~N}\) is now applied to the prism at B . This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each \(\frac { 1 } { 2 }\).

1. Show that the least value of \(P\) needed to move the prism is given by $$P = \frac { 40 \cos \theta + 95 \sin \theta } { 16 \sin \theta - 13 \cos \theta } .$$
2. Determine the range in which the value of \(\theta\) must lie.

12 Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u _ { 1 } \mathbf { i } + u _ { 2 } \mathbf { j }\) towards B , which is at rest.
The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf { i }\). The coefficient of restitution between A and B is \(e\).

1. 1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf { i }\).
   2. Determine this impulse in terms of \(\lambda , m , e\) and \(u _ { 1 }\). The loss in kinetic energy due to the collision between A and B is \(\frac { 1 } { 8 } \mathrm { mu } _ { 1 } { } ^ { 2 }\).
2. Determine the range of possible values of \(\lambda\).

13 A particle P of mass \(m\) is fixed to one end of a light spring of natural length \(a\) and modulus of elasticity man \({ } ^ { 2 }\), where \(n > 0\). The other end of the spring is attached to the ceiling of a lift. The lift is at rest and P is hanging vertically in equilibrium.

1. Find, in terms of \(g\) and \(n\), the extension in the spring. At time \(t = 0\) the lift begins to accelerate upwards from rest. At time \(t\), the upward displacement of the lift from its initial position is \(y\) and the extension of the spring is \(x\).
2. Express, in terms of \(g , n , x\) and \(y\), the upward displacement of P from its initial position at time \(t\).
3. Given that \(\ddot { y } = k t\), where \(k\) is a positive constant, express the upward acceleration of \(P\) in terms of \(\ddot { x } , k\) and \(t\).
4. Show that \(x\) satisfies the differential equation $$\ddot { \mathrm { x } } + \mathrm { n } ^ { 2 } \mathrm { x } = \mathrm { kt } + \mathrm { g } .$$
5. Verify that \(\mathrm { x } = \frac { 1 } { \mathrm { n } ^ { 3 } } ( \mathrm { k } n \mathrm { t } + \mathrm { gn } - \mathrm { k } \sin ( \mathrm { nt } ) )\).
6. By considering \(\dot { x }\) comment on the motion of P relative to the ceiling of the lift for all times after the lift begins to move.