3.03v Motion on rough surface: including inclined planes

4. A bullet of mass 50 g is fired horizontally at a wooden block of mass 4.95 kg which is lying at rest on a rough horizontal surface. The bullet enters the block at \(400 \mathrm {~ms} ^ { - 1 }\) and becomes embedded in the block.

1. Find the speed with which the block begins to move. Given that the block decelerates uniformly to rest over a distance of 4 m ,
2. show that the coefficient of friction is \(\frac { 2 } { g }\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a particle of mass 4 kg resting on the surface of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is connected by a light inextensible string passing over a smooth pulley at the top of the plane, to a particle of mass 5 kg which hangs freely. The coefficient of friction between the 4 kg mass and the plane is \(\mu\) and when the system is released from rest the 4 kg mass starts to move up the slope.

1. Show that the acceleration of the system is \(\frac { 1 } { 9 } ( 3 - 2 \mu \sqrt { 3 } ) \mathrm { g } \mathrm { ms } ^ { - 2 }\).
2. Hence, find the maximum value of \(\mu\). Given that \(\mu = \frac { 1 } { 2 }\),
3. find the tension in the string in terms of \(g\),
4. show that the magnitude of the force on the pulley is given by \(\frac { 5 } { 3 } ( 2 \sqrt { 3 } + 1 ) \mathrm { g }\). END

3. A cannon of mass 600 kg lies on a rough horizontal surface and is used to fire a 3 kg shell horizontally at \(200 \mathrm {~ms} ^ { - 1 }\).

1. Find the impulse which the shell exerts on the cannon.
2. Find the speed with which the cannon recoils. Given that the coefficient of friction between the cannon and the surface is 0.75 ,
3. calculate, to the nearest centimetre, the distance that the cannon travels before coming to rest.

2. A monk uses a small brush to clean the stone floor of a monastery by pushing the brush with a force of \(P\) Newtons at an angle of \(60 ^ { \circ }\) to the vertical. He moves the brush at a constant speed. The mass of the brush is 0.5 kg and the coefficient of friction between the brush and the floor is \(\frac { 1 } { \sqrt { 3 } }\). The brush is modelled as a particle and air resistance is ignored.

1. Show that \(P = \frac { g } { 2 }\) Newtons.
2. Explain why it is reasonable to ignore air resistance in this situation.

2 Robin is driving a car of mass 800 kg along a straight horizontal road at a speed of \(40 \mathrm {~ms} ^ { - 1 }\).
Robin applies the brakes and the car decelerates uniformly; it comes to rest after travelling a distance of 125 m .

1. Show that the resistance force on the car when the brakes are applied is 5120 N .
2. Find the time the car takes to come to rest. For the rest of this question, assume that when Robin applies the brakes there is a constant resistance force of 5120 N on the car. The car returns to its speed of \(40 \mathrm {~ms} ^ { - 1 }\) and the road remains straight and horizontal.
   Robin sees a red light 155 m ahead, takes a short time to react and then applies the brakes.
   The car comes to rest before it reaches the red light.
3. Show that Robin's reaction time is less than 0.75 s . The 'stopping distance' is the total distance travelled while a driver reacts and then applies the brakes to bring the car to rest. For the rest of this question, assume that Robin is still driving the car described above and has a reaction time of 0.675 s . (This is the figure used in calculating the stopping distances given in the Highway Code.)
4. Calculate the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) on a horizontal road. The car then travels down a hill which has a slope of \(5 ^ { \circ }\) to the horizontal.
5. Find the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) down this hill.
6. By what percentage is the stopping distance increased by the fact that the car is going down the hill? Give your answer to the nearest 1\%.

3 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
   The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_297_1180_971_741} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_351_1285_2038_466} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the PhysicsAptMaths, statter \&REther this rod is in tension or thrust (compression).

2 A train consists of a locomotive pulling 17 identical trucks.
The mass of the locomotive is 120 tonnes and the mass of each truck is 40 tonnes. The locomotive gives a driving force of 121000 N . The resistance to motion on each truck is \(R \mathrm {~N}\) and the resistance on the locomotive is \(5 R \mathrm {~N}\).
Initially the train is travelling on a straight horizontal track and its acceleration is \(0.11 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 1500\).
2. Find the tensions in the couplings between
   (A) the last two trucks,
   (B) the locomotive and the first truck. The train now comes to a place where the track goes up a straight, uniform slope at an angle \(\alpha\) with the horizontal, where \(\sin \alpha = \frac { 1 } { 80 }\). The driving force and the resistance forces remain the same as before.
3. Find the magnitude and direction of the acceleration of the train. The train then comes to a straight uniform downward slope at an angle \(\beta\) to the horizontal.
   The driver of the train reduces the driving force to zero and the resistance forces remain the same as before. The train then travels at a constant speed down the slope.
4. Find the value of \(\beta\).

2 The battery on Carol and Martin's car is flat so the car will not start. They hope to be able to "bump start" the car by letting it run down a hill and engaging the engine when the car is going fast enough. Fig. 6.1 shows the road leading away from their house, which is at A . The road is straight, and at all times the car is steered directly along it.

• From A to B the road is horizontal.
• Between B and C , it goes up a hill with a uniform slope of \(1.5 ^ { \circ }\) to the horizontal.
• Between C and D the road goes down a hill with a uniform slope of \(3 ^ { \circ }\) to the horizontal. CD is 100 m . (This is the part of the road where they hope to get the car started.)
• From D to E the road is again horizontal.

\begin{figure}[h] \end{figure} The mass of the car is 750 kg , Carol's mass is 50 kg and Martin's mass is 80 kg .
Throughout the rest of this question, whenever Martin pushes the car, he exerts a force of 300 N along the line of the car.

1. Between A and B, Martin pushes the car and Carol sits inside to steer it. The car has an acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that the resistance to the car's motion is 100 N . Throughout the rest of this question you should assume that the resistance to motion is constant at 100 N .
2. They stop at B and then Martin tries to push the car up the hill BC. Show that Martin cannot push the car up the hill with Carol inside it but can if she gets out.
   Find the acceleration of the car when Martin is pushing it and Carol is standing outside.
3. While between B and C, Carol opens the window of the car and pushes it from outside while steering with one hand. Carol is able to exert a force of 150 N parallel to the surface of the road but at an angle of \(30 ^ { \circ }\) to the line of the car. This is illustrated in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-2_216_425_1964_870} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} Find the acceleration of the car.
4. At C, both Martin and Carol get in the car and, starting from rest, let it run down the hill under gravity. If the car reaches a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) they can get the engine to start.

3 Fig. 7 illustrates a train with a locomotive, L, pulling two trucks, A and B. The locomotive has mass 90 tonnes and is subject to a resistance force of 2000 N .
Each of the trucks \(A\) and \(B\) has mass 30 tonnes and is subject to a resistance force of \(500 N\). \begin{figure}[h] \end{figure} Initially the train is travelling along a straight horizontal track. The locomotive is exerting a driving force of 12000 N .

1. Find the acceleration of the train.
2. Find the tension in the coupling between trucks A and B . When the train is travelling at \(10 \mathrm {~ms} ^ { - 1 }\), a fault occurs with truck A and the resistance to its motion changes from 500 N to 5000 N . The driver reduces the driving force to zero and allows the train to slow down under the resistance forces and come to a stop.
3. Find the distance the train travels while slowing down and coming to a stop. Find also the force in the coupling between trucks A and B while the train is slowing down, and state whether it is a tension or a thrust. The fault in truck A is repaired so that the resistance to its motion is again 500 N . The train continues and comes to a place where the track goes up a uniform slope at an angle of \(\alpha ^ { \circ }\) to the horizontal.
4. When the train is on the slope, it travels at uniform speed. The driving force remains at 12000 N . Find the value of \(\alpha\).
5. Show that the force in the coupling between trucks A and B has the same value that it had in part (ii).

5 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h] \end{figure} The sledge is initially at rest. After 10 seconds its speed is \(5 \mathrm {~ms} ^ { - 1 }\).

1. Show that the resistance to motion is 100 N . When the speed of the sledge is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks.
   The resistance to motion remains 100 N .
2. Find the speed of the sledge
   (A) 1.6 seconds after the rope breaks,
   (B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of \(15 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-5_263_854_1391_637} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
3. Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is \(8 \mathrm {~ms} ^ { - 1 }\), this rope also breaks.
4. Find the time between the rope breaking and the sledge reaching the bottom of the slope.

6 An empty open box of mass 4 kg is on a plane that is inclined at \(25 ^ { \circ }\) to the horizontal.
In one model the plane is taken to be smooth. The box is held in equilibrium by a string with tension \(T \mathrm {~N}\) parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h] \end{figure}

1. Calculate \(T\). A rock of mass \(m \mathrm {~kg}\) is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
2. Find \(m\). In a refined model the plane is rough. The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of \(P \mathrm {~N}\) inclined at \(15 ^ { \circ }\) to the plane, as shown in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_359_559_1599_830} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Draw a diagram showing all the forces acting on the box.
4. Calculate \(P\).
5. Calculate the normal reaction of the plane on the box.

5 A train consists of an engine of mass 10000 kg pulling one truck of mass 4000 kg . The coupling between the engine and the truck is light and parallel to the track. The train is accelerating at \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\) along a straight, level track.

1. What is the resultant force on the train in the direction of its motion? The driving force of the engine is 4000 N .
2. What is the resistance to the motion of the train?
3. If the tension in the coupling is 1150 N , what is the resistance to the motion of the truck? With the same overall resistance to motion, the train now climbs a uniform slope inclined at \(3 ^ { \circ }\) to the horizontal with the same acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\).
4. What extra driving force is being applied?

6 An elastic string has one end attached to a point \(O\), fixed on a horizontal table. The other end of the string is attached to a particle of mass 5 kilograms. The elastic string has natural length 2 metres and modulus of elasticity 200 newtons. The particle is pulled so that it is 2.5 metres from the point \(O\) and it is then released from rest on the table.

1. Calculate the elastic potential energy when the particle is 2.5 m from the point \(O\).
2. If the table is smooth, show that the speed of the particle when the string becomes slack is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The table is, in fact, rough and the coefficient of friction between the particle and the table is 0.4 . Find the speed of the particle when the string becomes slack.

8 Zoë carries out an experiment with a block, which she places on the horizontal surface of an ice rink. She attaches one end of a light elastic string to a fixed point, \(A\), on a vertical wall at the edge of the ice rink at the height of the surface of the ice rink. The block, of mass 0.4 kg , is attached to the other end of the string. The string has natural length 5 m and modulus of elasticity 120 N . The block is modelled as a particle which is placed on the surface of the ice rink at a point \(B\), where \(A B\) is perpendicular to the wall and of length 5.5 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-6_499_1429_813_333} The block is set into motion at the point \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) directly towards the point \(A\). The string remains horizontal throughout the motion.

1. Initially, Zoë assumes that the surface of the ice rink is smooth. Using this assumption, find the speed of the block when it reaches the point \(A\).
2. Zoë now assumes that friction acts on the block. The coefficient of friction between the block and the surface of the ice rink is \(\mu\).
   1. Find, in terms of \(g\) and \(\mu\), the speed of the block when it reaches the point \(A\).
   2. The block rebounds from the wall in the direction of the point \(B\). The speed of the block immediately after the rebound is half of the speed with which it hit the wall. Find \(\mu\) if the block comes to rest just as it reaches the point \(B\).

7. A cyclist is pedalling along a horizontal cycle track at a constant speed of \(5 \mathrm {~ms} ^ { - 1 }\). The air resistance opposing her motion has magnitude 42 N . The combined mass of the cyclist and her machine is 84 kg .

1. Find the rate, in W , at which the cyclist is working. The cyclist now starts to ascend a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\), at a constant speed.
   She continues to work at the same rate as before, against the same air resistance.
2. Find the constant speed at which she ascends the hill. In fact the air resistance is not constant, and a revised model takes account of this by assuming that the air resistance is proportional to the cyclist's speed.
3. Use this model to find an improved estimate of the speed at which she ascends the hill, if her rate of working still remains constant.

3 The resistance to the motion of a car of mass 600 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The car ascends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 10 }\). The power exerted by the car's engine is 12000 W and the car has constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 0.6\). The power exerted by the car's engine is increased to 16000 W .
2. Calculate the maximum speed of the car while ascending the hill. The car now travels on horizontal ground and the power remains 16000 W .
3. Calculate the acceleration of the car at an instant when its speed is \(32 \mathrm {~ms} ^ { - 1 }\).

2 A car of mass 1100 kg has maximum power of 44000 W . The resistive forces have constant magnitude 1400 N .

1. Calculate the maximum steady speed of the car on the level. The car is moving on a hill of constant inclination \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).
2. Calculate the maximum steady speed of the car when ascending the hill.
3. Calculate the acceleration of the car when it is descending the hill at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) working at half the maximum power.

1

1. Fig. 1.1 shows the velocities of a tanker of mass 120000 tonnes before and after it changed speed and direction. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-2_237_917_360_577} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} Calculate the magnitude of the impulse that acted on the tanker.
2. An object of negligible size is at rest on a horizontal surface. It explodes into two parts, P and Q , which then slide along the surface. Part P has mass 0.4 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Part Q has mass 0.5 kg .
   1. Calculate the speed of Q immediately after the explosion. State how the directions of motion of P and Q are related. The explosion takes place at a distance of 0.75 m from a raised vertical edge, as shown in Fig. 1.2. P travels along a line perpendicular to this edge. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-2_238_1205_1366_429} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure} After the explosion, P has a perfectly elastic direct collision with the raised edge and then collides again directly with Q . The collision between P and Q occurs \(\frac { 2 } { 3 } \mathrm {~s}\) after the explosion. Both collisions are instantaneous. The contact between P and the surface is smooth but there is a constant frictional force between Q and the surface.
   2. Show that Q has speed \(2.7 \mathrm {~ms} ^ { - 1 }\) just before P collides with it.
   3. Calculate the coefficient of friction between Q and the surface.
   4. Given that the coefficient of restitution between P and Q is \(\frac { 1 } { 8 }\), calculate the speed of Q immediately after its collision with P .

4 A rigid thin uniform rod AB with length 2.4 m and weight 30 N is used in different situations.

1. In the first situation, the rod rests on a small support 0.6 m from B and is held horizontally in equilibrium by a vertical string attached to A, as shown in Fig. 4.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_196_707_456_680} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Calculate the tension in the string and the force of the support on the rod.
2. In the second situation, the rod rests in equilibrium on the point of slipping with end A on a horizontal floor and the rod resting at P on a fixed block of height 0.9 m , as shown in Fig. 4.2. The rod is perpendicular to the edge of the block on which it rests and is inclined at \(\theta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_208_746_1101_657} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} (A) Suppose that the contact between the block and the rod is rough with coefficient of friction 0.6 and contact between the end A and the floor is smooth. Show that \(\tan \theta = 0.6\).
   (B) Suppose instead that the contact between the block and the rod is smooth and the contact between the end A and the floor is rough. The rod is now in limiting equilibrium at a different angle \(\theta\) such that the distance AP is 1.5 m . Calculate the normal reaction of the block on the rod. Calculate the coefficient of friction between the rod and the floor.

3

1. A car of mass 900 kg is travelling at a steady speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at arcsin 0.1 to the horizontal. The power required to do this is 20 kW . Calculate the resistance to the motion of the car.
2. A small box of mass 11 kg is placed on a uniform rough slope inclined at arc \(\cos \frac { 12 } { 13 }\) to the horizontal. The coefficient of friction between the box and the slope is \(\mu\).
   1. Show that if the box stays at rest then \(\mu \geqslant \frac { 5 } { 12 }\). For the remainder of this question, the box moves on a part of the slope where \(\mu = 0.2\).
      The box is projected up the slope from a point P with an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It travels a distance of 1.5 m along the slope before coming instantaneously to rest. During this motion, the work done against air resistance is 6 joules per metre.
   2. Calculate the value of \(v\). As the box slides back down the slope, it passes through its point of projection P and later reaches its initial speed at a point Q . During this motion, once again the work done against air resistance is 6 joules per metre.
   3. Calculate the distance PQ.

4 Jack and Jill are raising a pail of water vertically using a light inextensible rope. The pail and water have total mass 20 kg . In parts (i) and (ii), all non-gravitational resistances to motion may be neglected.

1. How much work is done to raise the pail from rest so that it is travelling upwards at \(0.5 \mathrm {~ms} ^ { - 1 }\) when at a distance of 4 m above its starting position?
2. What power is required to raise the pail at a steady speed of \(0.5 \mathrm {~ms} ^ { - 1 }\) ? Jack falls over and hurts himself. He then slides down a hill.
   His mass is 35 kg and his speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(3 \mathrm {~ms} ^ { - 1 }\) while descending through a vertical height of 3 m .
3. How much work is done against friction? In Jack's further motion, he slides down a slope at an angle \(\alpha\) to the horizontal where \(\sin \alpha = 0.1\). The frictional force on him is now constant at 150 N . For this part of the motion, Jack's initial speed is \(3 \mathrm {~ms} ^ { - 1 }\).
4. How much further does he slide before coming to rest?

4

1. A parachutist and her equipment have a combined mass of 80 kg . During a descent where the parachutist loses 1600 m in height, her speed reduces from \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she does \(1.3 \times 10 ^ { 6 } \mathrm {~J}\) of work against resistances. Use an energy method to calculate the value of \(V\).
2. A vehicle of mass 800 kg is climbing a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). At one time the vehicle has a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating up the hill at \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of 1150 N .
   1. Show that the driving force on the vehicle is 2134 N and calculate its power at this time. The vehicle is pulling a sledge, of mass 300 kg , which is sliding up the hill. The sledge is attached to the vehicle by a light, rigid coupling parallel to the slope. The force in the coupling is 900 N .
   2. Assuming that the only resistance to the motion of the sledge is due to friction, calculate the coefficient of friction between the sledge and the ground.

2 Fig. 2 shows a wedge of angle \(30 ^ { \circ }\) fixed to a horizontal floor. Small objects P , of mass 8 kg , and Q , of mass 10 kg , are connected by a light inextensible string that passes over a smooth pulley at the top of the wedge. The part of the string between P and the pulley is parallel to a line of greatest slope of the wedge. Q hangs freely and at no time does either P or Q reach the pulley or P reach the floor. \begin{figure}[h] \end{figure}

1. Assuming the string remains taut, find the change in the gravitational potential energy of the system when Q descends \(h \mathrm {~m}\), stating whether it is a loss or a gain. Object P makes smooth contact with the wedge. The system is set in motion with the string taut.
2. Find the speed at which Q hits the floor if
   (A) the system is released from rest with Q a distance of 1.2 m above the floor,
   (B) instead, the system is set in motion with Q a distance of 0.3 m above the floor and P travelling down the slope at \(1.05 \mathrm {~ms} ^ { - 1 }\). The sloping face is roughened so that the coefficient of friction between object P and the wedge is 0.9 . The system is set in motion with the string taut and P travelling down the slope at \(2 \mathrm {~ms} ^ { - 1 }\).
3. How far does P move before it reaches its lowest point?
4. Determine what happens to the system after P reaches its lowest point.
5. Calculate the power of the frictional force acting on P in part (iii) at the moment the system is set in motion. \section*{Question 3 begins on page 4.}

2

1. A bullet of mass 0.04 kg is fired into a fixed uniform rectangular block along a line through the centres of opposite parallel faces, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_209_1287_342_388} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} The bullet enters the block at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to rest after travelling 0.2 m into the block.
   1. Calculate the resistive force on the bullet, assuming that this force is constant. Another bullet of the same mass is fired, as before, with the same speed into a similar block of mass 3.96 kg . The block is initially at rest and is free to slide on a smooth horizontal plane.
   2. By considering linear momentum, find the speed of the block with the bullet embedded in it and at rest relative to the block.
   3. By considering mechanical energy, find the distance the bullet penetrates the block, given the resistance of the block to the motion of the bullet is the same as in part (i).
2. Fig. 2.2 shows a block of mass 6 kg on a uniformly rough plane that is inclined at \(30 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_348_636_1382_712} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A string with a constant tension of 91.5 N parallel to the plane pulls the block up a line of greatest slope. The speed of the block increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a distance of 8 m .