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.