3.03f Weight: W=mg

\includegraphics{figure_7} A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.

1. It is given that the sloping section of the slide is smooth.
   1. Find the speed of the child when she reaches the bottom of the sloping section. [3]
   2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest. [2]
2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide. [6]

Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at 30° to the horizontal, and \(A\) is higher up the plane than \(B\). When the particles collide, the speeds of \(A\) and \(B\) are 3 m s\(^{-1}\) and 2 m s\(^{-1}\) respectively. In the collision between the particles, the speed of \(A\) is reduced to 2.5 m s\(^{-1}\).

1. Find the speed of \(B\) immediately after the collision. [2]

After the collision, when \(B\) has moved 1.6 m down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. \(B\) hits the barrier 0.4 s after the collision, and when it hits the barrier, its speed is reduced by 90%. The two particles collide again 0.44 s after their previous collision, and they then coalesce on impact.

1. Show that the speed of \(B\) immediately after it hits the barrier is 0.5 m s\(^{-1}\). Hence find the speed of the combined particle immediately after the second collision between \(A\) and \(B\). [7]

A car has mass \(1250 \text{ kg}\).

1. The car is moving along a straight level road at a constant speed of \(36 \text{ m s}^{-1}\) and is subject to a constant resistance of magnitude \(850 \text{ N}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The hill is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta° = 0.1\), and the engine is working at \(63 \text{ kW}\). Find the speed of the car. [3]
3. The car descends the same hill with the engine of the car working at a constant rate of \(20 \text{ kW}\). The resistance is not constant. The initial speed of the car is \(20 \text{ m s}^{-1}\). Eight seconds later the car has speed \(24 \text{ m s}^{-1}\) and has moved \(176 \text{ m}\) down the hill. Use an energy method to find the total work done against the resistance during the eight seconds. [5]

A particle of mass \(3\text{ kg}\) is on a rough plane inclined at an angle of \(20°\) to the horizontal. A force of magnitude \(P\text{ N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is \(0.35\). Show that the least possible value of \(P\) is \(0.394\), correct to 3 significant figures, and find the greatest possible value of \(P\). [6]

A constant resistance to motion of magnitude 350 N acts on a car of mass 1250 kg. The engine of the car exerts a constant driving force of 1200 N. The car travels along a road inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.05\). Find the speed of the car when it has moved 100 m from rest in each of the following cases. • The car is moving up the hill. • The car is moving down the hill. [7]

\includegraphics{figure_3} A particle of mass \(0.6\) kg is placed on a rough plane which is inclined at an angle of \(21°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.3\). Show that the least possible value of \(P\) is \(0.470\), correct to \(3\) significant figures, and find the greatest possible value of \(P\). [6]

\includegraphics{figure_3} Two children, Alan and Bhavana, are standing on the horizontal floor of a lift, as shown in Figure 3. The lift has mass 250 kg. The lift is raised vertically upwards with constant acceleration by a vertical cable which is attached to the top of the lift. The cable is modelled as being light and inextensible. While the lift is accelerating upwards, the tension in the cable is 3616 N. As the lift accelerates upwards, the floor of the lift exerts a force of magnitude 565 N on Alan and a force of magnitude 226 N on Bhavana. Air resistance is modelled as being negligible and Alan and Bhavana are modelled as particles.

1. By considering the forces acting on the lift only, find the acceleration of the lift. [3]
2. Find the mass of Alan. [3]

A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find

1. the acceleration of the particle, [3]
2. the coefficient of friction between the particle and the plane. [5]

The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane. \includegraphics{figure_3}

1. Find the value of \(X\). [7]

A lifeboat slides down a straight ramp inclined at an angle of \(15°\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from rest at the top of the ramp and is moving with a speed of 12.6 m s\(^{-1}\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp. [9]

\includegraphics{figure_2} A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20°\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find

1. the normal reaction of the plane on the box, [3]
2. the acceleration of the box. [5]

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]

\includegraphics{figure_1} A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. A force of magnitude \(P\) newtons is applied to the box at 50° to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of \(P\), giving your answer to 2 significant figures. [9]

A woman travels in a lift. The mass of the woman is 50 kg and the mass of the lift is 950 kg. The lift is being raised vertically by a vertical cable which is attached to the top of the lift. The lift is moving upwards and has constant deceleration of \(2 \text{ m s}^{-2}\). By modelling the cable as being light and inextensible, find

1. the tension in the cable; [3]
2. the magnitude of the force exerted on the woman by the floor of the lift. [3]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(α\) to the horizontal, where \(\sin α = \frac{1}{16}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is operating at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]

1. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A van of mass 1500 kg is driving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{12}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is working at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]
2. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A cyclist and her bicycle have a total mass of \(70\) kg. She cycles along a straight horizontal road with constant speed \(3.5 \text{ ms}^{-1}\). She is working at a constant rate of \(490\) W.

1. Find the magnitude of the resistance to motion. [4]

The cyclist now cycles down a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), at a constant speed \(U \text{ ms}^{-1}\). The magnitude of the non-gravitational resistance to motion is modelled as \(40U\) newtons. She is now working at a constant rate of \(24\) W.

1. Find the value of \(U\). [7]

A cyclist and her cycle have a combined mass of \(75\) kg. The cyclist is cycling up a straight road inclined at \(5°\) to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude \(20\) N. At the instant when the cyclist has a speed of \(12\) m s\(^{-1}\), she is decelerating at \(0.2\) m s\(^{-2}\).

1. Find the rate at which the cyclist is working at this instant. [5]

When the cyclist passes the point \(A\) her speed is \(8\) m s\(^{-1}\). At \(A\) she stops working but does not apply the brakes. She comes to rest at the point \(B\). The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude \(20\) N.

1. Use the work-energy principle to find the distance \(AB\). [5]

A packing-case, of mass \(60\) kg, is standing on the floor of a lift. The mass of the lift-cage is \(200\) kg. The lift-cage is raised and lowered by means of a cable attached to its roof. In each of the following cases, find the magnitude of the force exerted by the floor of the lift-cage on the packing-case and the tension in the cable supporting the lift:

1. The lift is descending with constant speed. [3 marks]
2. The lift is ascending and accelerating at \(1.2 \text{ ms}^{-2}\). [4 marks]
3. State any modelling assumptions you have made. [2 marks]

A particle of mass \(0.1\) kg is at rest at a point \(A\) on a rough plane inclined at \(15°\) to the horizontal. The particle is given an initial velocity of \(6\) m s\(^{-1}\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after \(1.5\) s.

1. Find the coefficient of friction between the particle and the plane. [7]
2. Show that, after coming to instantaneous rest, the particle moves down the plane. [2]
3. Find the speed with which the particle passes through \(A\) during its downward motion. [6]

A particle \(P\) of mass \(0.5\) kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is \(0.6\).

1. Show that the magnitude of the frictional force acting on \(P\) is \(2.25\) N, correct to 3 significant figures. [3]
2. Find the acceleration of \(P\) when it is moving
   1. up the plane,
   2. down the plane.
   [4]
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4\) m s\(^{-1}\).
   1. Find the length of time before \(P\) reaches its highest point.
   2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
   [8]

A particle \(P\) of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is 0.6.

1. Show that the magnitude of the frictional force acting on \(P\) is 2.25 N, correct to 3 significant figures. [3]
2. Find the acceleration of \(P\) when it is moving
   1. up the plane,
   2. down the plane.
   [4]
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4 \text{ m s}^{-1}\).
   1. Find the length of time before \(P\) reaches its highest point.
   2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
   [8]

\includegraphics{figure_3} The diagram shows a small block \(B\), of mass \(3\) kg, and a particle \(P\), of mass \(0.8\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. \(B\) is held at rest on a horizontal surface, and \(P\) lies on a smooth plane inclined at \(30°\) to the horizontal. When \(B\) is released from rest it accelerates at \(0.2\) m s\(^{-2}\) towards the pulley.

1. By considering the motion of \(P\), show that the tension in the string is \(3.76\) N. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]

A block \(B\) of weight \(10\) N is projected down a line of greatest slope of a plane inclined at an angle of \(20°\) to the horizontal. \(B\) travels down the plane at constant speed.

1. 1. Find the components perpendicular and parallel to the plane of the contact force between \(B\) and the plane. [2]
   2. Hence show that the coefficient of friction is \(0.364\), correct to \(3\) significant figures. [2]
2. \includegraphics{figure_6} \(B\) is in limiting equilibrium when acted on by a force of \(T\) N directed towards the plane at an angle of \(45°\) to a line of greatest slope (see diagram). Given that the frictional force on \(B\) acts down the plane, find \(T\). [7]

A cyclist and her bicycle have a combined mass of 78 kg. While riding on level ground and using her greatest driving force, she is able to accelerate uniformly from rest to 10 ms\(^{-1}\) in 15 seconds against constant resistive forces that total 60 N.

1. Show that her maximum driving force is 112 N. [4 marks]

The cyclist begins to ascend a hill, inclined at an angle \(\alpha\) to the horizontal, riding with her maximum driving force and against the same resistive forces. In this case, she is able to maintain a steady speed.

1. Find the angle \(\alpha\), giving your answer to the nearest degree. [4 marks]
2. Comment on the assumption that the resistive force remains constant
   1. in the case when the cyclist is accelerating,
   2. in the case when she is maintaining a steady speed. [2 marks]

A car of mass 1.25 tonnes tows a caravan of mass 0.75 tonnes along a straight, level road. The total resistance to motion experienced by the car and the caravan is 1200 N. The car and caravan accelerate uniformly from rest to 25 m s\(^{-1}\) in 20 seconds.

1. Calculate the driving force produced by the car's engine. [4 marks]

Given that the resistance to motion experienced by the car and by the caravan are in the same ratio as their masses,

1. find these resistances and the tension in the towbar. [4 marks]

When the car and caravan are travelling at a steady speed of 25 m s\(^{-1}\), the towbar snaps. Assuming that the caravan experiences the same resistive force as before,

1. calculate the distance travelled by the caravan before it comes to rest, [5 marks]
2. give a reason why your answer to \((c)\) may be unrealistic. [2 marks]