3.03v Motion on rough surface: including inclined planes

\includegraphics{figure_6} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30°\) to the horizontal. \(P\) has mass \(0.2\text{ kg}\) and is held at rest on the plane. \(Q\) has mass \(0.2\text{ kg}\) and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is \(0.4\). The particle \(P\) is released.

1. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released. [6]

\(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of \(0.8\text{ m}\) before it comes to rest. \(P\) does not reach the pulley.

1. Find the speed of the particles immediately before \(Q\) strikes the floor. [5]
2. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion. [3]

A sledge of mass 25 kg is on a plane inclined at \(30°\) to the horizontal. The coefficient of friction between the sledge and the plane is 0.2.

1. \includegraphics{figure_6} The sledge is pulled up the plane, with constant acceleration, by means of a light cable which is parallel to a line of greatest slope (see Fig. 1). The sledge starts from rest and acquires a speed of \(0.8 \text{ m s}^{-1}\) after being pulled for 10 s. Ignoring air resistance, find the tension in the cable. [6]
2. \includegraphics{figure_7} On a subsequent occasion the cable is not in use and two people of total mass 150 kg are seated in the sledge. The sledge is held at rest by a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. Find the least value of \(P\) which will prevent the sledge from sliding down the plane. [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]

\includegraphics{figure_3} Figure 3 shows a block of mass 25 kg held in equilibrium on a plane inclined at an angle of 35° to the horizontal by means of a string which is at an angle of 15° to the line of greatest slope of the plane. In an initial model of the situation, the plane is assumed to be smooth. Giving your answers correct to 3 significant figures,

1. show that the tension in the string is 145 N. [3 marks]
2. find the magnitude of the reaction between the plane and the block. [4 marks]

In a more refined model, the plane is assumed to be rough. Given that the tension in the string can be increased to 200 N before the block begins to move up the slope,

1. find, correct to 3 significant figures, the magnitude of the frictional force and state the direction in which it acts. [4 marks]
2. Without performing any further calculations, state whether the reaction calculated in part (b) will increase, decrease or remain the same in the refined model. Give a reason for your answer. [3 marks]

\includegraphics{figure_4} Figure 4 shows two golf balls \(P\) and \(Q\) being held at the top of planes inclined at \(30°\) and \(60°\) to the vertical respectively. Both planes slope down to a common hole at \(H\), which is 3 m vertically below \(P\) and \(Q\). \(P\) is released from rest and travels down the line of greatest slope of the plane it is on which is assumed to be smooth.

1. Find the acceleration of \(P\) down the slope. [3 marks]
2. Show that the time taken for \(P\) to reach the hole is 0.904 seconds, correct to 3 significant figures. [5 marks] \(Q\) travels down the line of greatest slope of the plane it is on which is rough. The coefficient of friction between \(Q\) and the plane is \(\mu\). Given that the acceleration of \(Q\) down the slope is \(3 \text{ m s}^{-2}\),
3. find, correct to 3 significant figures, the value of \(\mu\). [5 marks] In order for the two balls to arrive at the hole at the same time, \(Q\) must be released \(t\) seconds before \(P\).
4. Find the value of \(t\) correct to 2 decimal places. [4 marks]

An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20°\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg. The coefficient of friction between the particle and the plane is 0.8. The three points, \(A\), \(B\) and \(C\), lie on a line of greatest slope. The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane. \includegraphics{figure_8}

1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N, correct to three significant figures. [3 marks]
2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
   1. Find \(x\). [8 marks]
   2. Find the acceleration of the particle as it starts to move back towards \(A\). [4 marks]

A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres. The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane. \includegraphics{figure_8} The particle is attached to two light elastic strings, \(PQ\) and \(PR\). One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N. The particle is released from rest at the point \(A\). The coefficient of friction between the particle and the plane is \(0.4\). Find the distance of the particle from \(Q\) when it is next at rest. [8 marks]

A van of mass 1600 kg is moving with constant speed down a straight road inclined at 7° to the horizontal. The non-gravitational resistance to the van's motion has a constant magnitude of 2000 N and the engine of the van is working at a rate of 1.5 kW. Find

1. the constant speed of the van, [5 marks]
2. the acceleration of the van if the resistance is suddenly reduced to 1900 N. [2 marks]

A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 N s directed up a line of greatest slope of the plane.

1. Find the speed at which the particle starts to move. [2]

The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).

1. Given that the plane is smooth, find the speed of the particle at \(Q\). [4]

It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed 3 ms\(^{-1}\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).

1. Given that the plane is inclined at 30° to the horizontal, find the magnitude of the frictional force on the particle. [4]

A small ball of mass 0.2 kg is projected with speed \(11 \text{ ms}^{-1}\) up a line of greatest slope of a roof from a point \(A\) at the bottom of the roof. The ball remains in contact with the roof and moves up the line of greatest slope to the top of the roof at \(B\). The roof is rough and the coefficient of friction is \(\frac{1}{4}\). The distance \(AB\) is 5 m and \(AB\) is inclined at \(30°\) to the horizontal (see diagram).

1. Show that the speed of the ball when it reaches \(B\) is \(5.44 \text{ ms}^{-1}\), correct to 2 decimal places. [6]

The ball leaves the roof at \(B\) and moves freely under gravity. The point \(C\) is at the lower edge of the roof. The distance \(BC\) is 5 m and \(BC\) is inclined at \(30°\) to the horizontal.

1. Determine whether or not the ball hits the roof between \(B\) and \(C\). [7]

\(A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(55°\) to the horizontal. \(A\) is below the level of \(B\) and \(AB = 4\) 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 \(6.7 \text{ m s}^{-1}\). The coefficient of friction between the plane and \(P\) is 0.15. Find

1. the work done against the frictional force as \(P\) moves from \(A\) to \(B\), [3]
2. the initial speed of \(P\) at \(A\). [4]

Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.8\). The coefficient of friction between A and the slope is 0.85. \includegraphics{figure_1}

1. Calculate the maximum possible frictional force between A and the slope. Show that A will remain at rest. [6]

With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has a speed of 16 m s\(^{-1}\) when it collides with A. In this collision the coefficient of restitution is 0.4, the impulses are parallel to the slope and linear momentum parallel to the slope is conserved.

1. Show that the velocity of A immediately after the collision is 8.4 m s\(^{-1}\) down the slope. Find the velocity of B immediately after the collision. [6]
2. Calculate the impulse on B in the collision. [3]

The blocks do not collide again.

1. For what length of time after the collision does A slide before it comes to rest? [4]

A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after \(2.4\) seconds. The horizontal component of the initial velocity of \(P\) is \(\frac{5}{3}d \text{ m s}^{-1}\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground. [1]
2. Find the vertical component of the initial velocity of \(P\). [2]

\(P\) just clears a vertical wall which is situated at a horizontal distance \(d\) m from \(O\).

1. Find the height of the wall. [3]

The speed of \(P\) as it passes over the wall is \(16 \text{ m s}^{-1}\).

1. Find the value of \(d\) correct to \(3\) significant figures. [4]

\includegraphics{figure_9} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg. The other end of the string is attached to a second particle \(B\) of mass 2.5 kg. Particle \(A\) is in contact with a rough plane inclined at \(\theta\) to the horizontal, where \(\cos \theta = \frac{4}{5}\). The string is taut and passes over a small smooth pulley \(P\) at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Particle \(B\) hangs freely below \(P\) at a distance 1.5 m above horizontal ground, as shown in the diagram. The coefficient of friction between \(A\) and the plane is \(\mu\). The system is released from rest and in the subsequent motion \(B\) hits the ground before \(A\) reaches \(P\). The speed of \(B\) at the instant that it hits the ground is \(1.2\) ms\(^{-1}\).

1. For the motion before \(B\) hits the ground, show that the acceleration of \(B\) is \(0.48\) ms\(^{-2}\). [1]
2. For the motion before \(B\) hits the ground, show that the tension in the string is \(23.3\) N. [3]
3. Determine the value of \(\mu\). [5] After \(B\) hits the ground, \(A\) continues to travel up the plane before coming to instantaneous rest before it reaches \(P\).
4. Determine the distance that \(A\) travels from the instant that \(B\) hits the ground until \(A\) comes to instantaneous rest. [4]

\includegraphics{figure_14} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg. The other end of the string is attached to a second particle \(B\) of mass 3 kg. Particle \(A\) is in contact with a smooth plane inclined at 30° to the horizontal and particle \(B\) is in contact with a rough horizontal plane. A second light inextensible string is attached to \(B\). The other end of this second string is attached to a third particle \(C\) of mass 4 kg. Particle \(C\) is in contact with a smooth plane \(\Pi\) inclined at an angle of 60° to the horizontal. Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from \(A\) to the pulley, and from \(C\) to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram). The coefficient of friction between \(B\) and the horizontal plane is \(\mu\). The system is released from rest and in the subsequent motion \(C\) moves down \(\Pi\) with acceleration \(a\) m s\(^{-2}\).

1. By considering an equation involving \(\mu\), \(a\) and \(g\) show that \(a < \frac{1}{9}g(2\sqrt{3} - 1)\). [7]
2. Given that \(a = \frac{1}{5}g\), determine the magnitude of the contact force between \(B\) and the horizontal plane. Give your answer correct to 3 significant figures. [4]

Block \(A\), of mass \(0.2\) kg, lies at rest on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, such that \(\tan \theta = \frac{7}{24}\) A light inextensible string is attached to \(A\) and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope. The other end of this string is attached to particle \(B\), of mass \(2\) kg, which is held at rest so that the string is taut, as shown in the diagram below. \includegraphics{figure_18}

1. \(B\) is released from rest so that it begins to move vertically downwards with an acceleration of \(\frac{543}{625}\) g ms\(^{-2}\) Show that the coefficient of friction between \(A\) and the surface of the inclined plane is \(0.17\) [8 marks]
2. In this question use \(g = 9.81\text{ ms}^{-2}\) When \(A\) reaches a speed of \(0.5\text{ ms}^{-1}\) the string breaks.
   1. Find the distance travelled by \(A\) after the string breaks until first coming to rest. [4 marks]
   2. State an assumption that could affect the validity of your answer to part (b)(i). [1 mark]

Two heavy boxes, \(M\) and \(N\), are connected securely by a length of rope. The mass of \(M\) is 50 kilograms. The mass of \(N\) is 80 kilograms. \(M\) is placed near the bottom of a rough slope. The slope is inclined at 60° above the horizontal. The rope is passed over a smooth pulley at the top end of the slope so that \(N\) hangs with the rope vertical. The boxes are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below. \includegraphics{figure_21} When the boxes are released, \(M\) moves up the slope as \(N\) descends, with acceleration \(a\) m s\(^{-2}\) The tension in the rope is \(T\) newtons.

1. Explain why the equation of motion for \(N\) is $$80g - T = 80a$$ [1 mark]
2. Show that the normal reaction force between \(M\) and the slope is \(25g\) newtons. [1 mark]
3. The coefficient of friction, \(\mu\), between the slope and \(M\) is such that \(0 \leq \mu \leq 1\) Show that $$a \geq \frac{(11 - 5\sqrt{3})g}{26}$$ [6 marks]
4. State one modelling assumption you have made throughout this question. [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.

1. Show that \(\mu = 0.83\) [4 marks]
2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
   1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
   2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]

One end of a light elastic string of natural length \(2.1\) m and modulus of elasticity \(4.8\) N is attached to a particle, \(P\), of mass \(1.75\) kg. The other end of the string is attached to a fixed point, \(O\), which is on a rough inclined plane. The angle between the plane and the horizontal is \(\theta\) where \(\sin\theta = \frac{3}{5}\). The coefficient of friction between \(P\) and the plane is \(0.732\). Particle \(P\) is placed on the plane at \(O\) and then projected down a line of greatest slope of the plane with an initial speed of \(2.4\) m s\(^{-1}\). Determine the distance that \(P\) has travelled from \(O\) at the instant when it first comes to rest. You can assume that during its motion \(P\) does not reach the bottom of the inclined plane. [8]

An object of mass 60 kg is on a rough plane inclined at an angle of 20° to the horizontal. The coefficient of friction between the object and the plane is \(0 \cdot 3\). Initially, the object is held at rest. A force which is parallel to the plane and of magnitude \(T\) N is applied to the object in an upward direction along the line of greatest slope. The object is then released.

1. Given that \(T = 15\), calculate the acceleration of the object down the plane. [6]
2. Given that \(T = 350\), determine whether or not the object moves up the plane. Give a reason for your answer. [3]

A car of mass 750 kg is moving on a slope inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = 0.1\). When the car's engine is working at a constant power \(PW\), the car can travel at maximum speeds of \(14\text{ ms}^{-1}\) up the slope and \(28\text{ ms}^{-1}\) down the slope. In each case, the resistance to motion experienced by the car is proportional to the square of its speed. Find the value of \(P\) and determine the resistance to the motion of the car when its speed is \(10.5\text{ ms}^{-1}\). [10]

A vehicle of mass 3500 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal. When the vehicle is travelling at a velocity of \(v\text{ ms}^{-1}\), the resistance to motion can be modelled by a variable force of magnitude \(40v\) N.

1. Given that \(\sin\alpha = \frac{3}{49}\), calculate the power developed by the engine at the instant when the speed of the vehicle is \(25\text{ ms}^{-1}\) and its deceleration is \(0.2\text{ ms}^{-2}\). [5]
2. When the vehicle's engine is working at a constant rate of 40 kW, the maximum speed that can be maintained up the slope is \(20\text{ ms}^{-1}\). Find the value of \(\alpha\). Give your answer in degrees, correct to one significant figure. [5]

A vehicle of mass 6000 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{6}{49}\). The vehicle's engine exerts a constant power of \(P\) W. The constant resistance to motion of the vehicle is \(R\) N. At the instant the vehicle is moving with velocity \(\frac{16}{5}\) ms\(^{-1}\), its acceleration is 2 ms\(^{-2}\). The maximum velocity of the vehicle is \(\frac{16}{3}\) ms\(^{-1}\). Determine the value of \(P\) and the value of \(R\). [9]

At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30°\) to the horizontal. The distance travelled down the chute by each brick is \(8\) m. A brick of mass \(3\) kg is released from rest at the top of the chute. When it reaches the bottom of the chute, its speed is \(5\) m s\(^{-1}\).

1. Find the potential energy lost by the brick in moving down the chute.

(2)

1. By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.

(5)

1. Hence find the coefficient of friction between the brick and the chute.

(3) Another brick of mass \(3\) kg slides down the chute. This brick is given an initial speed of \(2\) m s\(^{-1}\) at the top of the chute.

1. Find the speed of this brick when it reaches the bottom of the chute.

(5)

In this question use \(g = 10 \text{ m s}^{-2}\). A particle of mass 3 kg rests in limiting equilibrium on a rough plane inclined at \(30°\) to the horizontal.

1. Find the exact value of the coefficient of friction between the particle and the plane. [2]

A horizontal force of 36 N is now applied to the particle.

1. Find how far down the plane the particle travels after the force has been applied for 4 s. [6]