Questions - OCR M2 - A-Level Maths

OCR M2 2016 June Q1

1 A car of mass 1400 kg is travelling on a straight horizontal road against a constant resistance to motion of 600 N . At a certain instant the car is accelerating at $0.3 \mathrm {~ms} ^ { - 2 }$ and the engine of the car is working at a rate of 23 kW .

Find the speed of the car at this instant. Subsequently the car moves up a hill inclined at $10 ^ { \circ }$ to the horizontal at a steady speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to motion is still a constant 600 N .
Calculate the power of the car's engine as it moves up the hill.
$2 A$ and $B$ are two points on a line of greatest slope of a plane inclined at $55 ^ { \circ }$ to the horizontal. $A$ is below the level of $B$ and $A B = 4 \mathrm {~m}$. A particle $P$ of mass 2.5 kg is projected up the plane from $A$ towards $B$ and the speed of $P$ at $B$ is $6.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The coefficient of friction between the plane and $P$ is 0.15 . Find
the work done against the frictional force as $P$ moves from $A$ to $B$,
the initial speed of $P$ at $A$.

OCR M2 2016 June Q3

3 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-2_645_1024_1290_516} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A uniform lamina $A B D C$ is bounded by two semicircular arcs $A B$ and $C D$, each with centre $O$ and of radii $3 a$ and $a$ respectively, and two straight edges, $A C$ and $D B$, which lie on the line $A O B$ (see Fig. 1).

Show that the distance of the centre of mass of the lamina from $O$ is $\frac { 13 a } { 3 \pi }$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-3_1306_572_207_751} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The lamina has mass 3 kg and is freely pivoted to a fixed point at $A$. The lamina is held in equilibrium with $A B$ vertical by means of a light string attached to $B$. The string lies in the same plane as the lamina and is at an angle of $40 ^ { \circ }$ below the horizontal (see Fig. 2).
Calculate the tension in the string.
Find the direction of the force acting on the lamina at $A$.
\includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-4_848_1491_251_287} A smooth solid cone of semi-vertical angle $60 ^ { \circ }$ is fixed to the ground with its axis vertical. 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 vertically above the vertex of the cone. $P$ rotates in a horizontal circle on the surface of the cone with constant angular velocity $\omega$. The string is inclined to the downward vertical at an angle of $30 ^ { \circ }$ (see diagram).
Show that the magnitude of the contact force between the cone and the particle is $\frac { 1 } { 6 } m \left( 2 \sqrt { 3 } g - 3 a \omega ^ { 2 } \right)$.
Given that $a = 0.5 \mathrm {~m}$ and $m = 3.5 \mathrm {~kg}$, find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string.

OCR M2 2016 June Q5

5 A uniform ladder $A B$, of weight $W$ and length $2 a$, rests with the end $A$ in contact with rough horizontal ground and the end $B$ resting against a smooth vertical wall. The ladder is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 12 } { 13 }$. A man of weight 6 W is standing on the ladder at a distance $x$ from $A$ and the system is in equilibrium.

Show that the magnitude of the frictional force exerted by the ground on the ladder is $\frac { 5 W } { 24 } \left( 1 + \frac { 6 x } { a } \right)$. The coefficient of friction between the ladder and the ground is $\frac { 1 } { 3 }$.
Find, in terms of $a$, the greatest value of $x$ for which the system is in equilibrium. The bottom of the ladder $A$ is moved closer to the wall so that the ladder is now inclined at an angle $\alpha$ to the horizontal. The man of weight 6 W can now stand at the top of the ladder $B$ without the ladder slipping.
Find the least possible value of $\tan \alpha$.

OCR M2 2016 June Q6

6 The masses of two particles $A$ and $B$ are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. $A$ has speed $8 \mathrm {~ms} ^ { - 1 }$ and $B$ has speed $10 \mathrm {~ms} ^ { - 1 }$ before they collide. The kinetic energy lost due to the collision is 121.5 J .

Find the speed and direction of motion of each particle after the collision.
Find the coefficient of restitution between $A$ and $B$.
\includegraphics[max width=\textwidth, alt={}, center]{907be5c2-8a3c-482a-84fd-b45e8c36caa7-5_510_1504_653_271} A particle $P$ is projected with speed $32 \mathrm {~ms} ^ { - 1 }$ at an angle of elevation $\alpha$, where $\sin \alpha = \frac { 3 } { 5 }$, from a point $A$ on horizontal ground. At the same instant a particle $Q$ is projected with speed $20 \mathrm {~ms} ^ { - 1 }$ at an angle of elevation $\beta$, where $\sin \beta = \frac { 24 } { 25 }$, from a point $B$ on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point $C$ at the instant when they are travelling horizontally (see diagram).
Calculate the height of $C$ above the ground and the distance $A B$. Immediately after the collision $P$ falls vertically. $P$ hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
Given that the mass of $P$ is 3 kg , find the magnitude and direction of the impulse exerted on $P$ by the ground. The coefficient of restitution between the two particles is $\frac { 1 } { 2 }$.
Find the distance of $Q$ from $C$ at the instant when $Q$ is travelling in a direction of $25 ^ { \circ }$ below the horizontal.

OCR M2 Specimen Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_236_949_269_603} A barge $B$ is pulled along a canal by a horse $H$, which is on the tow-path. The barge and the horse move in parallel straight lines and the tow-rope makes a constant angle of $15 ^ { \circ }$ with the direction of motion (see diagram). The tow-rope remains taut and horizontal, and has a constant tension of 500 N .

Find the work done on the barge by the tow-rope, as the barge travels a distance of 400 m . The barge moves at a constant speed and takes 10 minutes to travel the 400 m .
Find the power applied to the barge.

OCR M2 Specimen Q2

2 A uniform circular cylinder, of radius 6 cm and height 15 cm , is in equilibrium on a fixed inclined plane with one of its ends in contact with the plane.

Given that the cylinder is on the point of toppling, find the angle the plane makes with the horizontal. The cylinder is now placed on a horizontal board with one of its ends in contact with the board. The board is then tilted so that the angle it makes with the horizontal gradually increases.
Given that the coefficient of friction between the cylinder and the board is $\frac { 3 } { 4 }$, determine whether or not the cylinder will slide before it topples, justifying your answer.

OCR M2 Specimen Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_389_698_1706_694} A uniform lamina $A B C D$ has the shape of a square of side $a$ adjoining a right-angled isosceles triangle whose equal sides are also of length $a$. The weight of the lamina is $W$. The lamina rests, in a vertical plane, on smooth supports at $A$ and $D$, with $A D$ horizontal (see diagram).

Show that the centre of mass of the lamina is at a horizontal distance of $\frac { 11 } { 9 } a$ from $A$.
Find, in terms of $W$, the magnitudes of the forces on the supports at $A$ and $D$.

OCR M2 Specimen Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-3_563_707_274_721} A rigid body $A B C$ consists of two uniform rods $A B$ and $B C$, rigidly joined at $B$. The lengths of $A B$ and $B C$ are 13 cm and 20 cm respectively, and their weights are 13 N and 20 N respectively. The distance of $B$ from $A C$ is 12 cm . The body hangs in equilibrium, with $A C$ horizontal, from two vertical strings attached at $A$ and $C$. Find the tension in each string.

OCR M2 Specimen Q5

5 A cyclist and his machine have a combined mass of 80 kg . The cyclist ascends a straight hill $A B$ of constant slope, starting from rest at $A$ and reaching a speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $B$. The level of $B$ is 4 m above the level of $A$.

Find the gain in kinetic energy and the gain in gravitational potential energy of the cyclist and his machine. During the ascent the resistance to motion is constant and has magnitude 70 N .
Given that the work done by the cyclist in ascending the hill is 8000 J , find the distance $A B$. At $B$ the cyclist is working at 720 watts and starts to move in a straight line along horizontal ground. The resistance to motion has the same magnitude of 70 N as before.
Find the acceleration with which the cyclist starts to move horizontally.

OCR M2 Specimen Q6

6 An athlete 'puts the shot' with an initial speed of $19 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $11 ^ { \circ }$ above the horizontal. At the instant of release the shot is 1.53 m above the horizontal ground. By treating the shot as a particle and ignoring air resistance, find

the maximum height, above the ground, reached by the shot,
the horizontal distance the shot has travelled when it hits the ground.

OCR M2 Specimen Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-4_314_757_285_708} A ball of mass 0.08 kg is attached by two strings to a fixed vertical post. The strings have lengths 2.5 m and 2.4 m , as shown in the diagram. The ball moves in a horizontal circle, of radius 2.4 m , with constant speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Each string is taut and the lower string is horizontal. The modelling assumptions made are that both strings are light and inextensible, and that there is no air resistance.

Find the tension in each string when $v = 10.5$.
Find the least value of $v$ for which the lower string is taut.

OCR M2 Specimen Q8

8 Two uniform smooth spheres, $A$ and $B$, have the same radius. The mass of $A$ is 0.24 kg and the mass of $B$ is $m \mathrm {~kg}$. Sphere $A$ is travelling in a straight line on a horizontal table, with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when it collides directly with sphere $B$, which is at rest. As a result of the collision, sphere $A$ continues in the same direction with a speed of $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the magnitude of the impulse exerted by $A$ on $B$.
Show that $m \leqslant 0.08$. It is given that $m = 0.06$.
Find the coefficient of restitution between $A$ and $B$. On another occasion $A$ and $B$ are travelling towards each other, each with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide directly.
Find the speeds of $A$ and $B$ immediately after the collision.

OCR M2 2008 January Q6

Show that the tension in the string is 4.16 N , correct to 3 significant figures.
Calculate $\omega$.
(ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The lower part of the string is now attached to a point $R$, vertically below $P$. $P B$ makes an angle $30 ^ { \circ }$ with the vertical and $R B$ makes an angle $60 ^ { \circ }$ with the vertical. The bead $B$ now moves in a horizontal circle of radius 1.5 m with constant speed $v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }$ (see Fig. 2).
Calculate the tension in the string.
Calculate $v$.

OCR M2 2006 June Q6

Calculate the tension in the string and hence find the angular speed of $Q$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle $Q$ on the plane is now fixed to a point 0.2 m from the hole at $A$ and the particle $P$ rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
Calculate the tension in the string.
Calculate the speed of $P$.

OCR M2 2008 June Q5

Show that the distance from the ball to the centre of mass of the toy is 10.7 cm , correct to 1 decimal place.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_312_1051_1509_587} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The toy lies on horizontal ground in a position such that the ball is touching the ground (see Fig. 2). Determine whether the toy is lying in equilibrium or whether it will move to a position where the rod is vertical.

OCR M2 2009 June Q5

Fig. 1 Fig. 1 shows a uniform lamina $B C D$ in the shape of a quarter circle of radius 6 cm . Show that the distance of the centre of mass of the lamina from $B$ is 3.60 cm , correct to 3 significant figures. A uniform rectangular lamina $A B D E$ has dimensions $A B = 12 \mathrm {~cm}$ and $A E = 6 \mathrm {~cm}$. A single plane object is formed by attaching the rectangular lamina to the lamina $B C D$ along $B D$ (see Fig. 2). The mass of $A B D E$ is 3 kg and the mass of $B C D$ is 2 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-3_959_447_1123_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
Taking $x$ - and $y$-axes along $A E$ and $A B$ respectively, find the coordinates of the centre of mass of the object. The object is freely suspended at $C$ and rests in equilibrium.
Calculate the angle that $A C$ makes with the vertical.

OCR M2 2015 June Q7

Show that $\mu = \frac { 2 } { 3 }$. A small object of weight $a W \mathrm {~N}$ is placed on the ladder at its mid-point and the object $S$ of weight $2 W \mathrm {~N}$ is placed on the ladder at its lowest point $A$.
Given that the system is in equilibrium, find the set of possible values of $a$.

OCR M2 Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-02_538_535_269_806} A uniform solid cone has vertical height 20 cm and base radius $r \mathrm {~cm}$. It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of inclination is $24 ^ { \circ }$ (see diagram).

Find $r$, correct to 1 decimal place. A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is inclined at an angle of $24 ^ { \circ }$.
State, with justification, whether this cone will topple.

OCR M2 Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-02_451_533_1676_808} One end of a light inextensible string of length 1.6 m is attached to a point $P$. The other end is attached to the point $Q$, vertically below $P$, where $P Q = 0.8 \mathrm {~m}$. A small smooth bead $B$, of mass 0.01 kg , is threaded on the string and moves in a horizontal circle, with centre $Q$ and radius $0.6 \mathrm {~m} . Q B$ rotates with constant angular speed $\omega$ rad s $^ { - 1 }$ (see diagram).

Show that the tension in the string is 0.1225 N .
Find $\omega$.
Calculate the kinetic energy of the bead.

OCR M2 Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-03_168_959_246_593} Three smooth spheres $A , B$ and $C$, of equal radius and of masses $m \mathrm {~kg} , 2 m \mathrm {~kg}$ and $3 m \mathrm {~kg}$ respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere $A$ is moving with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it collides directly with sphere $B$ which is stationary. As a result of the collision $B$ starts to move with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the coefficient of restitution between $A$ and $B$.
Find, in terms of $m$, the magnitude of the impulse that $A$ exerts on $B$, and state the direction of this impulse. Sphere $B$ subsequently collides with sphere $C$ which is stationary. As a result of this impact $B$ and $C$ coalesce.
Show that there will be another collision.

OCR M2 Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-03_319_650_1219_749} A uniform $\operatorname { rod } A B$ of length 60 cm and weight 15 N is freely suspended from its end $A$. The end $B$ of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point $C$ which is at the same horizontal level as $A$. The rod is in equilibrium with the string at right angles to the rod (see diagram).

Show that the tension in the string is 4.5 N .
Find the magnitude and direction of the force acting on the rod at $A$.

OCR M2 Q6

6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of $5 ^ { \circ }$ to the horizontal. At a certain point $P$ on the hill the car's speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The point $Q$ is 400 m further up the hill from $P$, and at $Q$ the car's speed is $15 \mathrm {~ms} ^ { - 1 }$.

Calculate the work done by the car's engine as the car moves from $P$ to $Q$, assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between $P$ and $Q$ is a constant force of magnitude 200 N.
Given that the acceleration of the car at $Q$ is zero, show that the power of the engine as the car passes through $Q$ is 12.0 kW , correct to 3 significant figures.
Given that the power of the car's engine at $P$ is the same as at $Q$, calculate the car's retardation at $P$. \section*{June 2005}

OCR M2 Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-04_397_1431_264_358} A barrier is modelled as a uniform rectangular plank of wood, $A B C D$, rigidly joined to a uniform square metal plate, $D E F G$. The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that $C D E$ is a straight line (see diagram). The barrier is smoothly pivoted at the point $D$. In the closed position, the barrier rests on a thin post at $H$. The distance $C H$ is 0.25 m .

Calculate the contact force at $H$ when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above $D$.
Calculate the angle between $A B$ and the horizontal when the barrier is in the open position.

OCR M2 Q8

17 marks

8 A particle is projected with speed $49 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation $\theta$ from a point $O$ on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from $O$ at time $t$ seconds after projection are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Express $x$ and $y$ in terms of $\theta$ and $t$, and hence show that $$y = x \tan \theta - \frac { x ^ { 2 } \left( 1 + \tan ^ { 2 } \theta \right) } { 490 } .$$
\includegraphics[max width=\textwidth, alt={}]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-04_638_1259_1695_443}
The particle passes through the point where $x = 70$ and $y = 30$. The two possible values of $\theta$ are $\theta _ { 1 }$ and $\theta _ { 2 }$, and the corresponding points where the particle returns to the plane are $A _ { 1 }$ and $A _ { 2 }$ respectively (see diagram).
Find $\theta _ { 1 }$ and $\theta _ { 2 }$.
Calculate the distance between $A _ { 1 }$ and $A _ { 2 }$.
Jan 2006 1 A uniform rod $A B$ has weight 20 N and length 3 m . The end $A$ is freely hinged to a point on a vertical wall. The rod is held horizontally and in equilibrium by a light inextensible string. One end of the string is attached to the rod at $B$. The other end of the string is attached to a point $C$, which is 1 m directly above $A$ (see diagram). Calculate the tension in the string. 2 A golfer hits a ball from a point $O$ on horizontal ground with a velocity of $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $25 ^ { \circ }$ above the horizontal. The ball first hits the ground at a point $A$. Assuming that there is no air resistance, calculate
the time taken for the ball to travel from $O$ to $A$,
the distance $O A$. 3 A box of mass 50 kg is dragged along a horizontal floor by a constant force of magnitude 400 N acting at an angle of $\alpha$ above the horizontal. The total resistance to the motion of the box has magnitude 300 N . The box starts from rest at the point $O$, and passes the point $P , 25 \mathrm {~m}$ from $O$, with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
For the box's motion from $O$ to $P$, find
(a) the increase in kinetic energy of the box,
(b) the work done against the resistance to motion of the box.
Hence calculate $\alpha$. 4
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-06_490_753_264_696} A rectangular frame consists of four uniform metal rods. $A B$ and $C D$ are vertical and each is 40 cm long and has mass $0.2 \mathrm {~kg} . A D$ and $B C$ are horizontal and each is 60 cm long. $A D$ has mass 0.7 kg and $B C$ has mass 0.5 kg . The frame is freely hinged at $E$ and $F$, where $E$ is 10 cm above $A$, and $F$ is 10 cm below $B$ (see diagram).
Sketch a diagram showing the directions of the horizontal components of the forces acting on the frame at $E$ and $F$.
Calculate the magnitude of the horizontal component of the force acting on the frame at $E$.
Calculate the distance from $A D$ of the centre of mass of the frame. 5 Three smooth spheres $A , B$ and $C$, of equal radius and of masses $3 m \mathrm {~kg} , 2 m \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively, are free to move in a straight line on a smooth horizontal table. Spheres $B$ and $C$ are stationary. Sphere $A$ is moving with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it collides directly with sphere $B$. The collision is perfectly elastic.
Find the velocities of $A$ and $B$ after the collision.
Find, in terms of $m$, the magnitude of the impulse that $A$ exerts on $B$, and state the direction of this impulse. Sphere $B$ continues its motion and hits $C$. After the collision, $B$ continues in the same direction with speed $1.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $C$ moves with speed $2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the coefficient of restitution between $B$ and $C$. 6 A stone is projected horizontally with speed $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ on the edge of a vertical cliff. The horizontal and upward vertical displacements of the stone from $O$ at any subsequent time, $t$ seconds, are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively. Assume that there is no air resistance.
Express $x$ and $y$ in terms of $t$, and hence show that $y = - \frac { 1 } { 10 } x ^ { 2 }$. The stone hits the sea at a point which is 20 m below the level of $O$.
Find the distance between the foot of the cliff and the point where the stone hits the sea.
Find the speed and direction of motion of the stone immediately before it hits the sea. Jan 2006
7 Marco is riding his bicycle at a constant speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a horizontal road, working at a constant rate of 300 W . Marco and his bicycle have a combined mass of 75 kg .
Calculate the wind resistance acting on Marco and his bicycle. Nicolas is riding his bicycle at the same speed as Marco and directly behind him. Nicolas experiences $30 \%$ less wind resistance than Marco.
Calculate the power output of Nicolas. The two cyclists arrive at the bottom of a hill which is at an angle of $1 ^ { \circ }$ to the horizontal. Marco increases his power output to 500 W .
Assuming Marco's wind resistance is unchanged, calculate his instantaneous acceleration immediately after starting to climb the hill. Marco reaches the top of the hill at a speed of $13 \mathrm {~ms} ^ { - 1 }$. He then freewheels down a hill of length 200 m which is at a constant angle of $10 ^ { \circ }$ to the horizontal. He experiences a constant wind resistance of 120 N .
Calculate Marco's speed at the bottom of this hill. 8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-08_282_711_264_717} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle $P$ of mass 0.1 kg is moving with constant angular speed $\omega \mathrm { rads } ^ { - 1 }$ in a horizontal circle on the smooth inner surface of a cone which is fixed with its axis vertical and its vertex $A$ at its lowest point. The semi-vertical angle of the cone is $60 ^ { \circ }$ and the distance $A P$ is 0.8 m (see Fig.1).
Calculate the magnitude of the force exerted by the cone on the particle.
Calculate $\omega$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-08_407_711_1103_717} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle $P$ is now attached to one end of a light inextensible string which passes through a small smooth hole at $A$. The lower end of the string is attached to a particle $Q$ of mass 0.2 kg . $Q$ is in equilibrium with the string taut and $A P = 0.8 \mathrm {~m} . P$ moves in a horizontal circle with constant speed $\nu \mathrm { m } \mathrm { s } ^ { - 1 }$ (see Fig. 2).
State the tension in the string.
Find $v$. \section*{June 2006} 1 A child of mass 35 kg runs up a flight of stairs in 10 seconds. The vertical distance climbed is 4 m . Assuming that the child's speed is constant, calculate the power output. 2 A small sphere of mass 0.3 kg is dropped from rest at a height of 2 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.4 m above the ground. Ignoring air resistance, calculate the magnitude of the impulse which the ground exerts on the sphere when it rebounds. 3
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-09_718_588_717_776} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point $O$, the centre of the plane face, and the other string is attached to the point $A$ on the rim of the plane face. The hemisphere hangs in equilibrium and $O A$ makes an angle of $60 ^ { \circ }$ with the vertical (see diagram).
Find the horizontal distance from the centre of mass of the hemisphere to the vertical through $O$.
Calculate the tensions in the strings. \section*{June 2006} 4 A car of mass 900 kg is travelling at a constant speed of $30 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a level road. The total resistance to motion is 450 N .
Calculate the power output of the car's engine. A roof box of mass 50 kg is mounted on the roof of the car. The total resistance to motion of the vehicle increases to 500 N .
The car's engine continues to work at the same rate. Calculate the maximum speed of the car on the level road. The power output of the car's engine increases to 15000 W . The resistance to motion of the car, with roof box, remains 500 N .
Calculate the instantaneous acceleration of the car on the level road when its speed is $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The car climbs a hill which is at an angle of $5 ^ { \circ }$ to the horizontal. Calculate the instantaneous retardation of the car when its speed is $26 \mathrm {~ms} ^ { - 1 }$. 5
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-10_668_565_1213_790} A uniform lamina $A B C D E$ consists of a square and an isosceles triangle. The square has sides of 18 cm and $B C = C D = 15 \mathrm {~cm}$ (see diagram).
Taking $x$ - and $y$-axes along $A E$ and $A B$ respectively, find the coordinates of the centre of mass of the lamina.
The lamina is freely suspended from $B$. Calculate the angle that $B D$ makes with the vertical. 6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-11_446_1358_262_391} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A light inextensible string of length 1 m passes through a small smooth hole $A$ in a fixed smooth horizontal plane. One end of the string is attached to a particle $P$, of mass 0.5 kg , which hangs in equilibrium below the plane. The other end of the string is attached to a particle $Q$, of mass 0.3 kg , which rotates with constant angular speed in a circle of radius 0.2 m on the surface of the plane (see Fig. 1).
Calculate the tension in the string and hence find the angular speed of $Q$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-11_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle $Q$ on the plane is now fixed to a point 0.2 m from the hole at $A$ and the particle $P$ rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
Calculate the tension in the string.
Calculate the speed of $P$. \section*{June 2006} 7 A small ball is projected at an angle of $50 ^ { \circ }$ above the horizontal, from a point $A$, which is 2 m above ground level. The highest point of the path of the ball is 15 m above the ground, which is horizontal. Air resistance may be ignored.
Find the speed with which the ball is projected from $A$. The ball hits a net at a point $B$ when it has travelled a horizontal distance of 45 m .
Find the height of $B$ above the ground.
Find the speed of the ball immediately before it hits the net. 8 Two uniform smooth spheres, $A$ and $B$, have the same radius. The mass of $A$ is 2 kg and the mass of $B$ is $m \mathrm {~kg}$. Sphere $A$ is travelling in a straight line on a smooth horizontal surface, with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when it collides directly with sphere $B$, which is at rest. As a result of the collision, sphere $A$ continues in the same direction with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the greatest possible value of $m$. It is given that $m = 1$.
Find the coefficient of restitution between $A$ and $B$. On another occasion $A$ and $B$ are travelling towards each other, each with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide directly.
Find the kinetic energy lost due to the collision. 1 A uniform solid cylinder has height 20 cm and diameter 12 cm . It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cylinder topples when the angle of inclination is $\alpha$. Find $\alpha$. 2 Two smooth spheres $A$ and $B$, of equal radius and of masses 0.2 kg and 0.1 kg respectively, are free to move on a smooth horizontal table. $A$ is moving with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it collides directly with $B$, which is stationary. The collision is perfectly elastic. Calculate the speed of $A$ after the impact. [4] 3 A small sphere of mass 0.2 kg is projected vertically downwards with speed $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point at a height of 40 m above horizontal ground. It hits the ground and rebounds vertically upwards, coming to instantaneous rest at its initial point of projection. Ignoring air resistance, calculate
the coefficient of restitution between the sphere and the ground,
the magnitude of the impulse which the ground exerts on the sphere. 4 A skier of mass 80 kg is pulled up a slope which makes an angle of $20 ^ { \circ }$ with the horizontal. The skier is subject to a constant frictional force of magnitude 70 N . The speed of the skier increases from $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the point $A$ to $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the point $B$, and the distance $A B$ is 25 m .
By modelling the skier as a small object, calculate the work done by the pulling force as the skier moves from $A$ to $B$.
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-13_458_1027_1420_598} It is given that the pulling force has constant magnitude $P \mathrm {~N}$, and that it acts at a constant angle of $30 ^ { \circ }$ above the slope (see diagram). Calculate $P$. 5 A model train has mass 100 kg . When the train is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the resistance to its motion is $3 v ^ { 2 } \mathrm {~N}$ and the power output of the train is $\frac { 3000 } { v } \mathrm {~W}$.
Show that the driving force acting on the train is 120 N at an instant when the train is moving with speed $5 \mathrm {~ms} ^ { - 1 }$.
Find the acceleration of the train at an instant when it is moving horizontally with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The train moves with constant speed up a straight hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 98 }$.
Calculate the speed of the train. 6
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-14_540_894_989_628} A uniform lamina $A B C D E$ of weight 30 N consists of a rectangle and a right-angled triangle. The dimensions are as shown in the diagram.
Taking $x$ - and $y$-axes along $A E$ and $A B$ respectively, find the coordinates of the centre of mass of the lamina. The lamina is freely suspended from a hinge at $B$.
Calculate the angle that $A B$ makes with the vertical. The lamina is now held in a position such that $B D$ is horizontal. This is achieved by means of a string attached to $D$ and to a fixed point 15 cm directly above the hinge at $B$.
Calculate the tension in the string. 7
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-15_787_1009_269_568} One end of a light inextensible string of length 0.8 m is attached to a fixed point $A$ which lies above a smooth horizontal table. The other end of the string is attached to a particle $P$, of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed $2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P$ makes an angle of $30 ^ { \circ }$ with the vertical (see diagram).
Calculate the tension in the string.
Calculate the normal contact force between the particle and the table. The particle now moves with constant speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is on the point of leaving the surface of the table.
Calculate $v$. 8 A missile is projected with initial speed $42 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $30 ^ { \circ }$ above the horizontal. Ignoring air resistance, calculate
the maximum height of the missile above the level of the point of projection,
the distance of the missile from the point of projection at the instant when it is moving downwards at an angle of $10 ^ { \circ }$ to the horizontal. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }\section*{June 2007} 1 A man drags a sack at constant speed in a straight line along horizontal ground by means of a rope attached to the sack. The rope makes an angle of $35 ^ { \circ }$ with the horizontal and the tension in the rope is 40 N . Calculate the work done in moving the sack 100 m . 2 Calculate the range on a horizontal plane of a small stone projected from a point on the plane with speed $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation of $27 ^ { \circ }$. 3 A rocket of mass 250 kg is moving in a straight line in space. There is no resistance to motion, and the mass of the rocket is assumed to be constant. With its motor working at a constant rate of 450 kW the rocket's speed increases from $100 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $150 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a time $t$ seconds.
Calculate the value of $t$.
Calculate the acceleration of the rocket at the instant when its speed is $120 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. 4 A ball is projected from a point $O$ on the edge of a vertical cliff. The horizontal and vertically upward components of the initial velocity are $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. At time $t$ seconds after projection the ball is at the point $( x , y )$ referred to horizontal and vertically upward axes through $O$. Air resistance may be neglected.
Express $x$ and $y$ in terms of $t$, and hence show that $y = 3 x - \frac { 1 } { 10 } x ^ { 2 }$. The ball hits the sea at a point which is 25 m below the level of $O$.
Find the horizontal distance between the cliff and the point where the ball hits the sea. 5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill $A B$ of constant slope, starting from rest at $A$ and reaching a speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $B$. The level of $B$ is 6 m above the level of $A$. For the cyclist's motion from $A$ to $B$, find
the increase in kinetic energy,
the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from $A$ to $B$ is 8000 J .
Calculate the distance $A B$. 6
\includegraphics[max width=\textwidth, alt={}, center]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-17_679_621_267_762} A particle $P$ of mass 0.3 kg is attached to one end of each of two light inextensible strings. The other end of the longer string is attached to a fixed point $A$ and the other end of the shorter string is attached to a fixed point $B$, which is vertically below $A$. $A P$ makes an angle of $30 ^ { \circ }$ with the vertical and is 0.4 m long. $P B$ makes an angle of $60 ^ { \circ }$ with the vertical. The particle moves in a horizontal circle with constant angular speed and with both strings taut (see diagram). The tension in the string $A P$ is 5 N . Calculate
the tension in the string $P B$,
the angular speed of $P$,
the kinetic energy of $P$. 7 Two small spheres $A$ and $B$, with masses 0.3 kg and $m \mathrm {~kg}$ respectively, lie at rest on a smooth horizontal surface. $A$ is projected directly towards $B$ with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and hits $B$. The direction of motion of $A$ is reversed in the collision. The speeds of $A$ and $B$ after the collision are $1 \mathrm {~ms} ^ { - 1 }$ and $3 \mathrm {~ms} ^ { - 1 }$ respectively. The coefficient of restitution between $A$ and $B$ is $e$.
Show that $m = 0.7$.
Find $e$.
$B$ continues to move at $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and strikes a vertical wall at right angles. The coefficient of restitution between $B$ and the wall is $f$.
Find the range of values of $f$ for which there will be a second collision between $A$ and $B$.
Find, in terms of $f$, the magnitude of the impulse that the wall exerts on $B$.
Given that $f = \frac { 3 } { 4 }$, calculate the final speeds of $A$ and $B$, correct to 1 decimal place. 8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-18_460_495_269_826} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} An object consists of a uniform solid hemisphere of weight 40 N and a uniform solid cylinder of weight 5 N . The cylinder has height $h \mathrm {~m}$. The solids have the same base radius 0.8 m and are joined so that the hemisphere's plane face coincides with one of the cylinder's faces. The centre of the common face is the point $O$ (see Fig.1). The centre of mass of the object lies inside the hemisphere and is at a distance of 0.2 m from $O$.
Show that $h = 1.2$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b2fd11a8-d5c3-4b90-93ff-a367ab5806de-18_630_1067_1292_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} One end of a light inextensible string is attached to a point on the circumference of the upper face of the cylinder. The string is horizontal and its other end is tied to a fixed point on a rough plane. The object rests in equilibrium on the plane with its axis of symmetry vertical. The plane makes an angle of $30 ^ { \circ }$ with the horizontal (see Fig. 2). The tension in the string is $T \mathrm {~N}$ and the frictional force acting on the object is $F \mathrm {~N}$.
By taking moments about $O$, express $F$ in terms of $T$.
Find another equation connecting $T$ and $F$. Hence calculate the tension and the frictional force.

Questions — OCR M2 (149 questions)

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