Questions - OCR M4 - A-Level Maths

OCR M4 2009 June Q1

1 A top is set spinning with initial angular speed $83 \mathrm { rad } \mathrm { s } ^ { - 1 }$, and it slows down with constant angular deceleration. When it has turned through 1000 radians, its angular speed is $67 \mathrm { rad } \mathrm { s } ^ { - 1 }$.

Find the angular deceleration of the top.
Find the time taken, from the start, for the top to turn through 400 radians.

OCR M4 2009 June Q2

2 The region $R$ is bounded by the $x$-axis, the lines $x = a$ and $x = 2 a$, and the curve $y = \frac { a ^ { 3 } } { x ^ { 2 } }$ for $a \leqslant x \leqslant 2 a$, where $a$ is a positive constant. A uniform solid of revolution is formed by rotating $R$ through $2 \pi$ radians about the $x$-axis. Find the $x$-coordinate of the centre of mass of this solid.

OCR M4 2009 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-2_664_623_904_760} A uniform circular disc has mass $4 m$, radius $2 a$ and centre $O$. The points $A$ and $B$ are at opposite ends of a diameter of the disc, and the mid-point of $O A$ is $P$. A particle of mass $m$ is attached to the disc at $B$. The resulting compound pendulum is in a vertical plane and is free to rotate about a fixed horizontal axis passing through $P$ and perpendicular to the disc (see diagram). The pendulum makes small oscillations.

Find the moment of inertia of the pendulum about the axis.
Find the approximate period of the small oscillations.

OCR M4 2009 June Q4

4 From a helicopter, a small plane is spotted 3750 m away on a bearing of $075 ^ { \circ }$. The plane is at the same altitude as the helicopter, and is flying with constant speed $62 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a horizontal straight line on a bearing of $295 ^ { \circ }$. The helicopter flies with constant speed $48 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a straight line, and intercepts the plane.

Find the bearings of the two possible directions in which the helicopter could fly.
Given that interception occurs in the shorter of the two possible times, find the time taken to make the interception.
\includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-3_668_298_260_922} A uniform lamina of mass 63 kg occupies the region bounded by the $x$-axis, the $y$-axis, and the curve $y = 8 - x ^ { 3 }$ for $0 \leqslant x \leqslant 2$. The unit of length is the metre. The vertices of the lamina are $O ( 0,0 )$, $A ( 2,0 )$ and $B ( 0,8 )$ (see diagram).
Show that the moment of inertia of this lamina about $O B$ is $56 \mathrm {~kg} \mathrm {~m} ^ { 2 }$. It is given that the moment of inertia of the lamina about $O A$ is $1036.8 \mathrm {~kg} \mathrm {~m} ^ { 2 }$, and the centre of mass of the lamina has coordinates $\left( \frac { 4 } { 5 } , \frac { 24 } { 7 } \right)$. The lamina is free to rotate in a vertical plane about a fixed horizontal axis passing through $O$ and perpendicular to the lamina. Starting with the lamina at rest with $B$ vertically above $O$, a couple of constant anticlockwise moment 800 Nm is applied to the lamina.
Show that the lamina begins to rotate anticlockwise.
Find the angular speed of the lamina at the instant when $O B$ first becomes horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-4_709_752_267_699} A smooth circular wire, with centre $O$ and radius $a$, is fixed in a vertical plane, and the point $A$ is on the wire at the same horizontal level as $O$. A small bead $B$ of mass $m$ can move freely on the wire. A light elastic string, with natural length $a$ and modulus of elasticity $\sqrt { 3 } m g$, passes through a fixed ring at $A$, and has one end fixed at $O$ and the other end attached to $B$. The section $A B$ of the string is at an angle $\theta$ above the horizontal, where $- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$, so that $O B$ is at an angle $2 \theta$ to the horizontal (see diagram).
Taking $O$ as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a ( \sqrt { 3 } + \sqrt { 3 } \cos 2 \theta + \sin 2 \theta ) .$$
Find the two values of $\theta$ for which the system is in equilibrium.
For each position of equilibrium, determine whether it is stable or unstable.
\includegraphics[max width=\textwidth, alt={}, center]{afecdb38-c372-480a-9d6d-fafe6a371dc2-5_478_1403_267_372} A thin horizontal rail is fixed at a height of 0.6 m above horizontal ground. A non-uniform straight $\operatorname { rod } A B$ has mass 6 kg and length 3 m ; its centre of mass $G$ is 2 m from $A$ and 1 m from $B$, and its moment of inertia about a perpendicular axis through its mid-point $M$ is $4.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }$. The rod is placed in a vertical plane perpendicular to the rail, with $A$ on the ground and $M$ in contact with the rail. It is released from rest in this position, and begins to rotate about $M$, without slipping on the rail. When the angle between $A B$ and the upward vertical is $\theta$ radians, the rod has angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$, the frictional force in the direction $A B$ is $F \mathrm {~N}$, and the normal reaction is $R \mathrm {~N}$ (see diagram).
Show that $\omega ^ { 2 } = 4.8 - 12 \cos \theta$.
Find the angular acceleration of the rod in terms of $\theta$.
Show that $F = 94.8 \cos \theta - 14.4$, and find $R$ in terms of $\theta$.
Given that the coefficient of friction between the rod and the rail is 0.9 , show that the rod will slip on the rail before $B$ hits the ground.

OCR M4 2010 June Q1

1 A wheel is rotating and is slowing down with constant angular deceleration. The initial angular speed is $80 \mathrm { rad } \mathrm { s } ^ { - 1 }$, and after 15 s the wheel has turned through 1020 radians.

Find the angular deceleration of the wheel.
Find the angle through which the wheel turns in the last 5 s before it comes to rest.
Find the total number of revolutions made by the wheel from the start until it comes to rest.

OCR M4 2010 June Q2

2 The region bounded by the $x$-axis, the $y$-axis, the line $x = \ln 3$, and the curve $y = \mathrm { e } ^ { - x }$ for $0 \leqslant x \leqslant \ln 3$, is occupied by a uniform lamina. Find, in an exact form, the coordinates of the centre of mass of this lamina.

OCR M4 2010 June Q3

3 A circular disc is rotating in a horizontal plane with angular speed $16 \mathrm { rad } \mathrm { s } ^ { - 1 }$ about a fixed vertical axis passing through its centre $O$. The moment of inertia of the disc about the axis is $0.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }$. A particle, initially at rest just above the surface of the disc, drops onto the disc and sticks to it at a point 0.4 m from $O$. Afterwards, the angular speed of the disc with the particle attached is $15 \mathrm { rad } \mathrm { s } ^ { - 1 }$.

Find the mass of the particle.
Find the loss of kinetic energy.

OCR M4 2010 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-2_688_777_1382_683} From a boat $B$, a cruiser $C$ is observed 3500 m away on a bearing of $040 ^ { \circ }$. The cruiser $C$ is travelling with constant speed $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a straight line course with bearing $110 ^ { \circ }$ (see diagram). The boat $B$ travels with constant speed $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a straight line course which takes it as close as possible to the cruiser $C$.

Show that the bearing of the course of $B$ is $073 ^ { \circ }$, correct to the nearest degree.
Find the magnitude and the bearing of the velocity of $C$ relative to $B$.
Find the shortest distance between $B$ and $C$ in the subsequent motion.

OCR M4 2010 June Q5

5 A uniform $\operatorname { rod } A B$ has mass $m$ and length $6 a$. The point $C$ on the rod is such that $A C = a$. The rod can rotate freely in a vertical plane about a fixed horizontal axis passing through $C$ and perpendicular to the rod.

Show by integration that the moment of inertia of the rod about this axis is $7 m a ^ { 2 }$. The rod starts at rest with $B$ vertically below $C$. A couple of constant moment $\frac { 6 m g a } { \pi }$ is then applied to the rod.
Find, in terms of $a$ and $g$, the angular speed of the rod when it has turned through one and a half revolutions.
\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-3_721_621_872_762} A light pulley of radius $a$ is free to rotate in a vertical plane about a fixed horizontal axis passing through its centre $O$. Two particles, $P$ of mass $5 m$ and $Q$ of mass $3 m$, are connected by a light inextensible string. The particle $P$ is attached to the circumference of the pulley, the string passes over the top of the pulley, and $Q$ hangs below the pulley on the opposite side to $P$. The section of string not in contact with the pulley is vertical. The fixed line $O X$ makes an angle $\alpha$ with the downward vertical, where $\cos \alpha = \frac { 4 } { 5 }$, and $O P$ makes an angle $\theta$ with $O X$ (see diagram). You are given that the total potential energy of the system (using a suitable reference level) is $V$, where $$V = m g a ( 3 \sin \theta - 4 \cos \theta - 3 \theta ) .$$

OCR M4 2010 June Q7

8 marks

7
\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-4_524_732_258_705} The diagram shows a uniform rectangular lamina $A B C D$ with $A B = 6 a , A D = 8 a$ and centre $G$. The mass of the lamina is $m$. The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through $A$ and perpendicular to the lamina.

Find the moment of inertia of the lamina about this axis. The lamina is released from rest with $A D$ horizontal and $B C$ below $A D$.
For an instant during the subsequent motion when $A D$ is vertical, show that the angular speed of the lamina is $\sqrt { \frac { 3 g } { 50 a } }$ and find its angular acceleration. At an instant when $A D$ is vertical, the force acting on the lamina at $A$ has magnitude $F$.
By finding components parallel and perpendicular to $G A$, or otherwise, show that $F = \frac { \sqrt { 493 } } { 20 } \mathrm { mg }$.
[0pt] [8]

OCR M4 2011 June Q1

1 When the power is turned off, a fan disk inside a jet engine slows down with constant angular deceleration $0.8 \mathrm { rad } \mathrm { s } ^ { - 2 }$.

Find the time taken for the angular speed to decrease from $950 \mathrm { rad } \mathrm { s } ^ { - 1 }$ to $750 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
Find the angle through which the disk turns as the angular speed decreases from $220 \mathrm { rad } \mathrm { s } ^ { - 1 }$ to $200 \mathrm { rad } \mathrm { s } ^ { - 1 }$.
Find the time taken for the disk to make the final 10 revolutions before coming to rest.

OCR M4 2011 June Q2

2 A straight $\operatorname { rod } A B$ has length $a$. The rod has variable density, and at a distance $x$ from $A$ its mass per unit length is $k \mathrm { e } ^ { - \frac { x } { a } }$, where $k$ is a constant. Find, in an exact form, the distance of the centre of mass of the rod from $A$.

OCR M4 2011 June Q3

3 A uniform rod $X Y$, of mass 5 kg and length 1.8 m , is free to rotate in a vertical plane about a fixed horizontal axis through $X$. The rod is at rest with $Y$ vertically below $X$ when a couple of constant moment is applied to the rod. It then rotates, and comes instantaneously to rest when $X Y$ is horizontal.

Find the moment of the couple.
Find the angular acceleration of the rod
(a) immediately after the couple is first applied,
(b) when $X Y$ is horizontal.

OCR M4 2011 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-2_439_1045_1512_550} Two small smooth pegs $A$ and $B$ are fixed at a distance $2 a$ apart on the same horizontal level, and $C$ is the mid-point of $A B$. A uniform rod $C D$, of mass $m$ and length $a$, is freely pivoted at $C$ and can rotate in the vertical plane containing $A B$, with $D$ below the level of $A B$. A light elastic string, of natural length $a$ and modulus of elasticity $3 m g$, passes round the peg $A$ and its ends are attached to $C$ and $D$. Another light elastic string, of natural length $a$ and modulus of elasticity $4 m g$, passes round the peg $B$ and its ends are also attached to $C$ and $D$. The angle $C A D$ is $\theta$, where $0 < \theta < \frac { 1 } { 2 } \pi$, so that the angle $B C D$ is $2 \theta$ (see diagram).

Taking $A B$ as the reference level for gravitational potential energy, show that the total potential energy of the system is $$\frac { 1 } { 2 } m g a ( 14 - 2 \cos 2 \theta - \sin 2 \theta )$$
Find the value of $\theta$ for which the system is in equilibrium.
Determine whether this position of equilibrium is stable or unstable.

OCR M4 2011 June Q5

5 The region inside the circle $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ is rotated about the $x$-axis to form a uniform solid sphere of radius $a$ and volume $\frac { 4 } { 3 } \pi a ^ { 3 }$. The mass of the sphere is $10 M$.

Show by integration that the moment of inertia of the sphere about the $x$-axis is $4 M a ^ { 2 }$. (You may assume the standard formula $\frac { 1 } { 2 } m r ^ { 2 }$ for the moment of inertia of a uniform disc about its axis.) The sphere is free to rotate about a fixed horizontal axis which is a diameter of the sphere. A particle of mass $M$ is attached to the lowest point of the sphere. The sphere with the particle attached then makes small oscillations as a compound pendulum.
Find, in terms of $a$ and $g$, the approximate period of these oscillations.

OCR M4 2011 June Q6

6 Two ships $P$ and $Q$ are moving on straight courses with constant speeds. At one instant $Q$ is 80 km from $P$ on a bearing of $220 ^ { \circ }$. Three hours later, $Q$ is 36 km due south of $P$.

Show that the velocity of $Q$ relative to $P$ is $19.1 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ in the direction with bearing $063.8 ^ { \circ }$ (both correct to 3 significant figures).
Find the shortest distance between the two ships in the subsequent motion. Given that the speed of $P$ is $28 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and $Q$ is travelling in the direction with bearing $105 ^ { \circ }$, find
the bearing of the direction in which $P$ is travelling,
the speed of $Q$.

OCR M4 2011 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-3_479_1225_1484_461} A uniform rectangular block of mass $m$ and cross-section $A B C D$ has $A B = C D = 6 a$ and $A D = B C = 2 a$. The point $X$ is on $A B$ such that $A X = a$ and $G$ is the centre of $A B C D$. The block is placed with $A B$ perpendicular to the straight edge of a rough horizontal table. $A X$ is in contact with the table and $X B$ overhangs the edge (see diagram). The block is released from rest in this position, and it rotates without slipping about a horizontal axis through $X$.

Find the moment of inertia of the block about the axis of rotation. For the instant when $X G$ is horizontal,
show that the angular acceleration of the block is $\frac { 3 \sqrt { 5 } g } { 25 a }$,
find the angular speed of the block,
show that the force exerted by the table on the block has magnitude $\frac { 2 \sqrt { 70 } } { 25 } m g$.

OCR M4 2012 June Q1

1 A uniform square lamina, of mass 4.5 kg and side 0.6 m , is rotating about a fixed vertical axis which is perpendicular to the lamina and passes through its centre. A stationary particle becomes attached to the lamina at one of its corners, and this causes the angular speed of the lamina to change instantaneously from $2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$ to $1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }$.

Find the mass of the particle. The lamina then slows down with constant angular deceleration. It turns through 36 radians as its angular speed reduces from $1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }$ to zero.
Find the time taken for the lamina to come to rest.

OCR M4 2012 June Q2

2 A uniform solid of revolution is formed by rotating the region bounded by the $x$-axis and the curve $y = x \left( 1 - \frac { x ^ { 2 } } { a ^ { 2 } } \right)$ for $0 \leqslant x \leqslant a$, where $a$ is a constant, about the $x$-axis. Find the $x$-coordinate of the centre of mass of this solid.

OCR M4 2012 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-2_460_388_1160_826} A ship $S$ is travelling with constant velocity $15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a course with bearing $120 ^ { \circ }$. A patrol boat $B$ observes the ship when $S$ is due north of $B$. The patrol boat $B$ then moves with constant speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a straight line (see diagram).

Given that $V = 18$, find the bearing of the course of $B$ such that $B$ intercepts $S$.
Given instead that $V = 9$, find the bearing of the course of $B$ such that $B$ passes as close as possible to $S$.
Find the smallest value of $V$ for which it is possible for $B$ to intercept $S$.

OCR M4 2012 June Q4

4 A uniform lamina of mass 18 kg occupies the region bounded by the $x$-axis, the $y$-axis, the line $x = \ln 9$ and the curve $y = \mathrm { e } ^ { \frac { 1 } { 2 } x }$ for $0 \leqslant x \leqslant \ln 9$. The unit of length is the metre. Find the moment of inertia of this lamina about the $x$-axis.

OCR M4 2012 June Q5

5 A uniform rod of mass 4 kg and length 2.4 m can rotate in a vertical plane about a fixed horizontal axis through one end of the rod. The rod is released from rest in a horizontal position and a frictional couple of constant moment 20 Nm opposes the motion.

Find the angular acceleration of the rod immediately after it is released.
Find the angle that the rod makes with the horizontal when its angular acceleration is zero.
Find the maximum angular speed of the rod.
The rod first comes to instantaneous rest after rotating through an angle $\theta$ radians from its initial position. Find an equation for $\theta$, and verify that $2.0 < \theta < 2.1$.

OCR M4 2012 June Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-3_716_483_890_790} Two small smooth pegs $P$ and $Q$ are fixed at a distance $2 a$ apart on the same horizontal level, and $A$ is the mid-point of $P Q$. A light rod $A B$ of length $4 a$ is freely pivoted at $A$ and can rotate in the vertical plane containing $P Q$, with $B$ below the level of $P Q$. A particle of mass $m$ is attached to the rod at $B$. A light elastic string, of natural length $2 a$ and modulus of elasticity $\lambda$, passes round the pegs $P$ and $Q$ and its two ends are attached to the rod at the point $X$, where $A X = a$. The angle between the rod and the downward vertical is $\theta$, where $- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$ (see diagram). You are given that the elastic energy stored in the string is $\lambda a ( 1 + \cos \theta )$.

Show that $\theta = 0$ is a position of equilibrium, and show that the equilibrium is stable if $\lambda < 4 m g$.
Given that $\lambda = 3 m g$, show that $\ddot { \theta } = - k \frac { g } { a } \sin \theta$, stating the value of the constant $k$. Hence find the approximate period of small oscillations of the system about the equilibrium position $\theta = 0$.

OCR M4 2012 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-4_783_783_255_641} A uniform circular disc with centre $C$ has mass $m$ and radius $a$. The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point $A$ on the disc, where $A C = \frac { 1 } { 2 } a$. The disc is slightly disturbed from rest in the position with $C$ vertically above $A$. When $A C$ makes an angle $\theta$ with the upward vertical the force exerted by the axis on the disc has components $R$ parallel to $A C$ and $S$ perpendicular to $A C$ (see diagram).

Show that the angular speed of the disc is $\sqrt { \frac { 4 g ( 1 - \cos \theta ) } { 3 a } }$.
Find the angular acceleration of the disc, in terms of $a , g$ and $\theta$.
Find $R$ and $S$, in terms of $m , g$ and $\theta$.
Find the magnitude of the force exerted by the axis on the disc at an instant when $R = 0$.

OCR M4 2013 June Q1

1 A camshaft inside an engine is rotating with angular speed $42 \mathrm { rads } ^ { - 1 }$. When the throttle is opened the camshaft speeds up with constant angular acceleration, and 8 seconds after the throttle was opened the angular speed is $76 \mathrm { rad } \mathrm { s } ^ { - 1 }$.

Find the angular acceleration of the camshaft.
Find the time taken for the camshaft to turn through 810 radians from the moment that the throttle was opened.

Questions — OCR M4 (100 questions)

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