Questions — OCR MEI Further Mechanics Major (73 questions)

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OCR MEI Further Mechanics Major 2023 June Q12
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. 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 \(\lambda\).
OCR MEI Further Mechanics Major 2023 June Q13
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.
OCR MEI Further Mechanics Major 2020 November Q1
1 A particle P of mass 0.5 kg is attached to a fixed point O by a light elastic string of natural length 3 m and modulus of elasticity 75 N . P is released from rest at O and is allowed to fall freely.
Determine the length of the string when P is at its lowest point in the subsequent motion.
OCR MEI Further Mechanics Major 2020 November Q2
2 A student conducts an experiment by first stretching a length of wire and fixing its ends. The student then plucks the wire causing it to vibrate. The frequency of these vibrations, \(f\), is modelled by the formula
\(f = k C ^ { \alpha } l ^ { \beta } \sigma ^ { \gamma }\),
where \(C\) is the tension in the wire,
\(l\) is the length of the stretched wire,
\(\sigma\) is the mass per unit length of the stretched wire and
\(k\) is a dimensionless constant. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
OCR MEI Further Mechanics Major 2020 November Q3
3 The vertices of a triangular lamina, which is in the \(x - y\) plane, are at the origin O and the points \(A ( 2,3 )\) and \(B ( - 2,1 )\). Forces \(2 \mathbf { i } + \mathbf { j }\) and \(- 3 \mathbf { i } + 2 \mathbf { j }\) are applied to the lamina at A and B , respectively, and a force \(\mathbf { F }\), whose line of action is in the \(x - y\) plane, is applied at O . The three forces form a couple.
  1. Determine the magnitude and the direction of \(\mathbf { F }\).
  2. Determine the magnitude and direction of the additional couple that must be applied to the lamina in order to keep it in equilibrium.
OCR MEI Further Mechanics Major 2020 November Q4
4 A particle P moves so that its position vector \(\mathbf { r }\) at time \(t\) is given by \(\mathbf { r } = ( 5 + 20 t ) \mathbf { i } + \left( 95 + 10 t - 5 t ^ { 2 } \right) \mathbf { j }\).
  1. Determine the initial velocity of P . At time \(t = T , \mathrm { P }\) is moving in a direction perpendicular to its initial direction of motion.
  2. Determine the value of \(T\).
  3. Determine the distance of P from its initial position at time \(T\).
OCR MEI Further Mechanics Major 2020 November Q5
5 A car of mass 900 kg moves along a straight level road. The power developed by the car is constant and equal to 60 kW . The resistance to the motion of the car is constant and equal to 1500 N . At time \(t\) seconds the velocity of the car is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Initially the car is at rest.
  1. Show that \(\frac { 3 v } { 5 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = 40 - v\).
  2. Verify that \(t = 24 \ln \left( \frac { 40 } { 40 - v } \right) - \frac { 3 } { 5 } v\).
OCR MEI Further Mechanics Major 2020 November Q6
6 A small ball of mass \(m \mathrm {~kg}\) is held at a height of 78.4 m above horizontal ground. The ball is released from rest, falls vertically and rebounds from the ground. The coefficient of restitution between the ball and ground is \(e\). The ball continues to bounce until it comes to rest after 6 seconds.
  1. Determine the value of \(e\).
  2. Given that the magnitude of the impulse that the ground exerts on the ball at the first bounce is 23.52 Ns , determine the value of \(m\).
OCR MEI Further Mechanics Major 2020 November Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-04_483_988_989_251} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} A particle P of mass \(m\) is attached to one end of a light elastic string of natural length \(6 a\) and modulus of elasticity 3 mg . The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A , as shown in Fig. 7. P is now pulled a further distance \(2 a\) down the line of greatest slope through A and released from rest. At time \(t\) later, the displacement of P from A is \(x\), where the positive direction of \(x\) is down the plane.
  1. Show that, until the string slackens, \(x\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + \frac { g x } { 2 a } = 0$$
  2. Determine, in terms of \(a\) and \(g\), the time at which the string slackens.
  3. Find, in terms of \(a\) and \(g\), the speed of P when the string slackens.
OCR MEI Further Mechanics Major 2020 November Q8
8 [In this question, you may use the fact that the volume of a right circular cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  1. By using integration, show that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac { 3 } { 4 } h\) from the vertex. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-05_929_679_504_333} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure} Fig. 8 shows the side view of a toy formed by joining a uniform solid circular cylinder of radius \(r\) and height \(2 r\) to a uniform solid right circular cone, made of the same material as the cylinder, of radius \(r\) and height \(r\). The toy is placed on a horizontal floor with the curved surface of the cone in contact with the floor.
  2. Determine whether the toy will topple.
  3. Explain why it is not necessary to know whether the floor is rough or smooth in answering part (b). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-06_397_1036_264_255} \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{figure} Fig. 9 shows a uniform rod AB of length \(2 a\) and weight \(8 W\) which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight \(W\) is threaded on the bar and is connected to the rod at B by a light inextensible string of length \(2 a\). The system is in equilibrium with the rod inclined at an angle \(\theta\) to the horizontal.
OCR MEI Further Mechanics Major 2020 November Q11
11 Two uniform small smooth spheres A and B have equal radii and equal masses. The spheres are on a smooth horizontal surface. Sphere A is moving at an acute angle \(\alpha\) to the line of centres, when it collides with B, which is stationary. After the impact A is moving at an acute angle \(\beta\) to the line of centres. The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\).
  1. Show that \(\tan \beta = 3 \tan \alpha\).
  2. Explain why the assumption that the contact between the spheres is smooth is needed in answering part (a). It is given that A is deflected through an angle \(\gamma\).
  3. Determine, in terms of \(\alpha\), an expression for \(\tan \gamma\).
  4. Determine the maximum value of \(\gamma\). You do not need to justify that this value is a maximum. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-09_488_903_264_258} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P , of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.
  5. Show that the normal contact force between P and the bowl is of magnitude \(m g + 2 m r \omega ^ { 2 } \cos ^ { 2 } \alpha\).
  6. Deduce that \(g < r \omega ^ { 2 } \left( k _ { 1 } + k _ { 2 } \cos ^ { 2 } \alpha \right)\), stating the value of the constants \(k _ { 1 }\) and \(k _ { 2 }\).
OCR MEI Further Mechanics Major 2021 November Q1
1 A small ball of mass 0.25 kg is held above a horizontal floor. The ball is released from rest and hits the floor with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It rebounds from the floor with a speed of \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The situation is modelled by assuming that the ball is in contact with the floor for 0.02 s and during this time the normal contact force the floor exerts on the ball is constant. Determine the magnitude of the normal contact force that the floor exerts on the ball.
OCR MEI Further Mechanics Major 2021 November Q2
2 The diagram shows a system of three particles of masses \(3 m , 5 m\) and \(2 m\) situated in the \(x - y\) plane at the points \(\mathrm { A } ( 1,2 ) , \mathrm { B } ( 2 , - 2 )\) and \(\mathrm { C } ( 5,3 )\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-02_789_744_1085_239} Determine the coordinates of the centre of mass of the system of particles.
OCR MEI Further Mechanics Major 2021 November Q3
3 One end of a light elastic spring of natural length 0.3 m is attached to a fixed point. A mass of 4 kg is attached to the other end of the spring. When the spring hangs vertically in equilibrium the extension of the spring is 0.02 m .
  1. Determine the modulus of elasticity of the spring. A student calculates that if the mass of 4 kg is removed and replaced with a mass of 20 kg the extension of the spring will be 0.1 m .
  2. Suggest a reason why this extension may not be 0.1 m .
OCR MEI Further Mechanics Major 2021 November Q4
4 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242} The diagram shows parts of the curves \(y = 3 \sqrt { x }\) and \(y = 4 - x ^ { 2 }\), which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the \(y\)-axis, is occupied by a uniform lamina. Determine the exact \(x\)-coordinate of the centre of mass of the lamina.
OCR MEI Further Mechanics Major 2021 November Q5
5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and B has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along a line which is perpendicular to the line of centres (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Determine the coefficient of restitution between A and B .
OCR MEI Further Mechanics Major 2021 November Q6
6
  1. Write down the dimensions of force. The force \(F\) of gravitational attraction between two objects with masses \(m _ { 1 }\) and \(m _ { 2 }\), at a distance \(d\) apart, is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is the universal gravitational constant.
    In SI units the value of \(G\) is \(6.67 \times 10 ^ { - 11 } \mathrm {~kg} ^ { - 1 } \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 2 }\).
  2. Write down the dimensions of \(G\).
  3. Determine the value of \(G\) in imperial units based on pounds, feet, and seconds. Use the facts that 1 pound \(= 0.454 \mathrm {~kg}\) and 1 foot \(= 0.305 \mathrm {~m}\). For a planet of mass \(M\) and radius \(r\), it is suggested that the velocity \(v\) needed for an object to escape the gravitational pull of the planet, the 'escape velocity', is given by the following formula.
    \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\),
    where \(k\) is a dimensionless constant.
  4. Show that this formula is dimensionally consistent. Information regarding the planets Earth and Mars can be found in the table below.
    EarthMars
    Radius (m)63710003389500
    Mass (kg)\(5.97 \times 10 ^ { 24 }\)\(6.39 \times 10 ^ { 23 }\)
    Escape velocity ( \(\mathrm { m } \mathrm { s } ^ { - 1 }\) )11186
  5. Using the formula \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\), determine the escape velocity for planet Mars.
OCR MEI Further Mechanics Major 2021 November Q7
7 A box B of mass \(m \mathrm {~kg}\) is raised vertically by an engine working at a constant rate of \(k m g \mathrm {~W}\). Initially B is at rest. The speed of B when it has been raised a distance \(x \mathrm {~m}\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } \frac { d v } { d x } = ( k - v ) g\).
  2. Verify that \(\mathrm { gx } = \mathrm { k } ^ { 2 } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { v } } \right) - \mathrm { kv } - \frac { 1 } { 2 } \mathrm { v } ^ { 2 }\).
  3. By using the work-energy principle, show that the time taken for B to reach a speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest is given by
    \(\frac { \mathrm { k } } { \mathrm { g } } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { V } } \right) - \frac { \mathrm { V } } { \mathrm { g } }\).
OCR MEI Further Mechanics Major 2021 November Q8
8 A capsule consists of a uniform hollow right circular cylinder of radius \(r\) and length \(2 h\) attached to two uniform hollow hemispheres of radius \(r\).
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17e92314-d7df-49b8-a441-8d18c91dbbb0-06_702_684_445_244} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall. The total weight of the capsule is \(W\) and acts at a point midway between A and B . The plane containing \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D is vertical, with AB making an acute angle \(\theta\) with the downward vertical.
  1. Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule. The coefficient of friction at each point of contact is \(\frac { 1 } { 3 }\).
  2. By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of \(W\).
  3. By determining an expression for \(r\) in terms of \(h\) and \(\theta\), show that \(\tan \theta > \frac { 3 } { 4 }\).
OCR MEI Further Mechanics Major 2021 November Q9
9 A small ball P is projected with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(( \alpha + \theta )\) from a point O at the bottom of a plane inclined at \(\alpha\) to the horizontal. P subsequently hits the plane at a point R , where OR is a line of greatest slope, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-07_456_862_406_242}
  1. By deriving an expression, in terms of \(\theta\), \(\alpha\) and \(g\), for the time of flight of P , show that the distance OR, in metres, is $$\frac { 50 \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
  2. By using the identity \(2 \sin \mathrm {~A} \cos \mathrm {~B} \equiv \sin ( \mathrm {~A} + \mathrm { B } ) - \sin ( \mathrm { B } - \mathrm { A } )\), determine, in terms of \(g\) and \(\sin \alpha\), an expression for the maximum range of P up the plane, as \(\theta\) varies.
  3. Given that OR is the maximum range of P up the plane and is equal to 1.8 m , determine the value of \(\theta\).
    \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-08_625_1180_255_239} A rigid wire ABC is fixed in a vertical plane. The section AB of the wire, of length \(b\), is straight and horizontal. The section BC of the wire is smooth and in the form of a circular arc of radius \(a\) and length \(\frac { 1 } { 2 } a \pi\). The centre of the arc is O , which is vertically above B . A bead P of mass \(m\) is threaded on the wire and projected from B with speed \(u\) towards C . The angle BOP when P is between B and C is denoted by \(\theta\), as shown in the diagram.
OCR MEI Further Mechanics Major 2021 November Q11
11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 2 } { 5 }\). At a given instant B is at rest and A is set moving along the groove with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It may be assumed that in the subsequent motion the two spheres do not leave the groove.
  1. Determine, in terms of \(e\) and \(V\), the speeds of A and B immediately after the first collision.
  2. Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
    1. Determine, in terms of \(e\) and \(V\), the speed of B immediately after the second collision.
    2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?
OCR MEI Further Mechanics Major 2021 November Q12
12 A particle P of mass \(m\) is fixed to one end of a light elastic string of natural length \(l\) and modulus of elasticity 12 mg . The other end of the string is attached to a fixed point O . Particle P is held next to O and then released from rest.
  1. Show that P next comes instantaneously to rest when the length of the string is \(\frac { 3 } { 2 } l\). The string first becomes taut at time \(t = 0\). At time \(t \geqslant 0\), the length of the string is \(l + x\), where \(x\) is the extension in the string.
  2. Show that when the string is taut, \(x\) satisfies the differential equation $$\ddot { \mathrm { x } } + \omega ^ { 2 } \mathrm { x } = \mathrm { g } \text {, where } \omega ^ { 2 } = \frac { 12 \mathrm {~g} } { \mathrm { I } } \text {. }$$
  3. By using the substitution \(x = y + \frac { g } { \omega ^ { 2 } }\), solve the differential equation to show that the time when the string first becomes slack satisfies the equation $$\cos \omega \mathrm { t } - \sqrt { \mathrm { k } } \sin \omega \mathrm { t } = 1$$ where \(k\) is an integer to be determined.
OCR MEI Further Mechanics Major 2023 June Q9
  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 \(\mathrm { u } ^ { 2 } > 2 \mathrm { gr }\).
OCR MEI Further Mechanics Major 2020 November Q9
  1. Determine, in terms of \(W\) and \(\theta\), the tension in the string. It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is \(2.4 a\).
  2. Determine the coefficient of friction between the bar and the ring. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-07_850_835_258_255} \captionsetup{labelformat=empty} \caption{Fig. 10}
    \end{figure} Fig. 10 shows a small bead P of mass \(m\) which is threaded on a smooth thin wire. The wire is in the form of a circle of radius \(a\) and centre O . The wire is fixed in a vertical plane. The bead is initially at the lowest point A of the wire and is projected along the wire with a velocity which is just sufficient to carry it to the highest point on the wire. The angle between OP and the downward vertical is denoted by \(\theta\).
  3. Determine the value of \(\theta\) when the magnitude of the reaction of the wire on the bead is \(\frac { 7 } { 2 } m g\).
  4. Show that the angular velocity of P when OP makes an angle \(\theta\) with the downward vertical is given by \(k \sqrt { \frac { g } { a } } \cos \left( \frac { \theta } { 2 } \right)\), stating the value of the constant \(k\).
  5. Hence determine, in terms of \(g\) and \(a\), the angular acceleration of P when \(\theta\) takes the value found in part (a).
OCR MEI Further Mechanics Major 2021 November Q10
  1. Determine the magnitude of the normal reaction of the wire on P in terms of \(m , g , a , u\) and \(\theta\), when P is between B and C . P collides with a fixed barrier at C . The coefficient of restitution between P and the fixed barrier is \(e\). After this collision P moves back towards B . On the straight portion BA , the motion of P is resisted by a constant horizontal force \(F\).
  2. Show that P will reach A if $$F b \leqslant \frac { 1 } { 2 } m \left[ e ^ { 2 } u ^ { 2 } + k \left( 1 - e ^ { 2 } \right) g a \right] ,$$ where \(k\) is an integer to be determined.