3.03v Motion on rough surface: including inclined planes

A particle of mass 0.8 kg is moving in a straight line on a rough horizontal plane. The speed of the particle is reduced from 15 m s\(^{-1}\) to 10 m s\(^{-1}\) as the particle moves 20 m. Assuming that the only resistance to motion is the friction between the particle and the plane, find

1. the work done by friction in reducing the speed of the particle from 15 m s\(^{-1}\) to 10 m s\(^{-1}\), [2]
2. the coefficient of friction between the particle and the plane. [4]

\includegraphics{figure_2} A uniform rod \(AB\), of mass \(20\) kg and length \(4\) m, rests with one end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\), by a force acting at \(B\), as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is \(0.5\). Find the magnitude of the normal reaction of the ground on the rod at \(A\). [7]

\includegraphics{figure_2} A uniform rod \(AB\) has mass \(4\) kg and length \(1.4\) m. The end \(A\) is resting on rough horizontal ground. A light string \(BC\) has one end attached to \(B\) and the other end attached to a fixed point \(C\). The string is perpendicular to the rod and lies in the same vertical plane as the rod. The rod is in equilibrium, inclined at \(20°\) to the ground, as shown in Figure 2.

1. Find the tension in the string. [4]

Given that the rod is about to slip,

1. find the coefficient of friction between the rod and the ground. [7]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.5. The other end \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of 30° with the wall. A man of mass \(5m\) stands on the ladder which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(ka\). Find the value of \(k\). [9]

A child is playing with a small model of a fire-engine of mass 0.5 kg and a straight, rigid plank. The plank is inclined at an angle \(\alpha\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons. When \(\alpha = 20°\) the fire-engine is projected with an initial speed of 5 m s\(^{-1}\) and first comes to rest after travelling 2 m.

1. Find, to 3 significant figures, the value of \(R\). [7]

When \(\alpha = 40°\) the fire-engine is again projected with an initial speed of 5 m s\(^{-1}\).

1. Find how far the fire-engine travels before first coming to rest. [3]

\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\). The cable makes an angle \(\beta\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is 0.6. The log is modelled as a uniform rod and the cable as light.

1. Show that the normal reaction on the log at \(A\) is \(\frac{5}{8}W\). [6]
2. Find the value of \(\beta\). [6]

The tension in the cable is \(kW\).

1. Find the value of \(k\). [2]

A particle \(P\) has mass 4 kg. It is projected from a point \(A\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\frac{2}{5}\). The particle comes to rest instantaneously at the point \(B\) on the plane, where \(AB = 2.5\) m. It then moves back down the plane to \(A\).

1. Find the work done by friction as \(P\) moves from \(A\) to \(B\). [4]

1. Using the work-energy principle, find the speed with which \(P\) is projected from \(A\). [4]

1. Find the speed of \(P\) when it returns to \(A\). [4]

A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at 30° to the horizontal. When \(P\) has moved 12 m, its speed is 4 m s\(^{-1}\). Given that friction is the only non-gravitational resistive force acting on \(P\), find

1. the work done against friction as the speed of \(P\) increases from 0 m s\(^{-1}\) to 4 m s\(^{-1}\), (4)
2. the coefficient of friction between the particle and the plane. (4)

\includegraphics{figure_2} A particle \(P\) of mass 0.5 kg is projected from a point \(A\) up a line of greatest slope \(AB\) of a fixed plane. The plane is inclined at 30° to the horizontal and \(AB = 2\) m with \(B\) above \(A\), as shown in Figure 2. The particle \(P\) passes through \(B\) with speed 5 m s\(^{-1}\). The plane is smooth from \(A\) to \(B\).

1. Find the speed of projection. [4]

The particle \(P\) comes to instantaneous rest at the point \(C\) on the plane, where \(C\) is above \(B\) and \(BC = 1.5\) m. From \(B\) to \(C\) the plane is rough and the coefficient of friction between \(P\) and the plane is \(\mu\). By using the work-energy principle,

1. find the value of \(\mu\). [6]

\includegraphics{figure_3} A uniform rod \(AB\) has weight 30 N and length 3 m. The rod rests in equilibrium on a rough horizontal peg \(P\) with its end \(A\) on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at 15° to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.

1. Show that the normal reaction at \(P\) has magnitude 25 N. [4]
2. Find the magnitude of the force on the rod at \(A\). [4]

The coefficient of friction between the rod and the peg is \(\mu\).

1. Find the range of possible values of \(\mu\). [4]

One end \(A\) of a light elastic string \(AB\), of modulus of elasticity \(mg\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(AB\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(AB = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.

1. Show that when the particle comes to rest it has moved a distance \(2a(\sin \theta - \mu \cos \theta)\) down the plane. [6]
2. Given that there is no further motion, show that \(\mu \geqslant \frac{1}{3} \tan \theta\). [5]

A force of magnitude \(F\) N is applied to a block of mass \(M\) kg which is initially at rest on a horizontal plane. The block starts to move with acceleration 3 ms\(^{-2}\). Modelling the block as a particle, \includegraphics{figure_4}

1. if the plane is smooth, find an expression for \(F\) in terms of \(M\). [2 marks]

If the plane is rough, and the coefficient of friction between the block and the plane is \(\mu\),

1. express \(F\) in terms of \(M\), \(\mu\) and \(g\). [2 marks]
2. Calculate the value of \(\mu\) if \(F = \frac{1}{2}Mg\). [3 marks]

A small package \(P\), of mass 1 kg, is initially at rest on the rough horizontal top surface of a wooden packing case which is 1.5 m long and 1 m high and stands on a horizontal floor. The coefficient of friction between \(P\) and the case is 0.2. \(P\) is attached by a light inextensible string, which passes over a smooth fixed pulley, to a weight \(Q\) of mass \(M\) kg which rests against the smooth vertical side of the case. The system is released from rest with \(P\) 0.75 m from the pulley and \(Q\) 0.5 m from the pulley. \(P\) and \(Q\) start to move with acceleration 0.4 ms\(^{-2}\). Calculate

1. the tension in the string, in N, [3 marks]
2. the value of \(M\), [3 marks]
3. the time taken for \(Q\) to hit the floor. [3 marks]

Given that \(Q\) does not rebound from the floor,

1. calculate the distance of \(P\) from the pulley when it comes to rest. [6 marks]

\includegraphics{figure_2}

\includegraphics{figure_3} A small packet, of mass \(1.2\) kg, is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the packet and the plane is \(\frac{1}{8}\). When a force of magnitude \(8.4\) N, acting parallel to the plane, is applied to the packet as shown, the packet is just on the point of moving up the plane. Modelling the packet as a particle,

1. show that \(7(\cos \alpha + 8 \sin \alpha) = 40\). \hfill [6 marks]

Given that the solution of this equation is \(\alpha = 38°\),

1. find the acceleration with which the packet moves down the plane when it is released from rest with no external force applied. \hfill [4 marks]

Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(m\) respectively, are connected by a light inextensible string which passes over a smooth fixed pulley as shown. \(A\) is initially at rest on the rough horizontal surface of a table, the coefficient of friction between \(A\) and the table being \(\frac{2}{7}\). \(B\) hangs freely on the end of the vertical portion of the string. \includegraphics{figure_5} \(A\) is now given an impulse, directed away from the pulley, of magnitude \(5m\) Ns.

1. Show that the system starts to move with speed \(2.5 \text{ ms}^{-1}\). [1 mark]
2. State which modelling assumption ensures that the tensions in the two sections of the string can be taken to be equal. [1 mark]

Given that \(A\) comes to rest before it reaches the edge of the table and before \(B\) hits the pulley,

1. find the time taken for the system to come to rest. [7 marks]
2. Find the distance travelled by \(A\) before it first comes to rest. [4 marks]

A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at 30° to the horizontal as shown. A light string is attached to \(P\) and makes an angle of 30° with the plane. When the tension in this string has magnitude \(kmg\), \(P\) is just on the point of moving up the plane. \includegraphics{figure_7}

1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac{k\sqrt{3} - 1}{\sqrt{3} - k}\). [7 marks]
2. Given further that \(k = \frac{3\sqrt{3}}{7}\), deduce that \(\mu = \frac{\sqrt{3}}{6}\). [3 marks]

The string is now removed.

1. Determine whether \(P\) will move down the plane and, if it does, find its acceleration. [5 marks]
2. Give a reason why the way in which \(P\) is shown in the diagram might not be consistent with the modelling assumptions that have been made. [1 mark]

A string is attached to a packing case of mass 12 kg, which is at rest on a rough horizontal plane. When a force of magnitude 50 N is applied at the other end of the string, and the string makes an angle of 35° with the vertical as shown, the case is on the point of moving. \includegraphics{figure_3}

1. Find the coefficient of friction between the case and the plane. [5 marks]

The force is now increased, with the string at the same angle, and the case starts to move along the plane with constant acceleration, reaching a speed of 2 ms\(^{-1}\) after 4 seconds.

1. Find the magnitude of the new force. [5 marks]
2. State any modelling assumptions you have made about the case and the string. [2 marks]

Two particles \(P\) and \(Q\), of masses \(2m\) and \(3m\) respectively, are connected by a light string. Initially, \(P\) is at rest on a smooth horizontal table. The string passes over a small smooth pulley and \(Q\) rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{4}{3}\). The coefficient of friction between \(Q\) and the inclined plane is \(\frac{1}{6}\). \includegraphics{figure_7} The system is released from rest with \(Q\) at a distance of 0.8 metres above a horizontal floor.

1. Show that the acceleration of \(P\) and \(Q\) is \(\frac{21g}{50}\), stating a modelling assumption which you must make to ensure that both particles have the same acceleration. [7 marks]
2. Find the speed with which \(Q\) hits the floor. [2 marks]

After \(Q\) hits the floor and does not rebound, \(P\) moves a further 0.2 m until it hits the pulley.

1. Find the total time after the system is released before \(P\) hits the pulley. [5 marks]

\includegraphics{figure_5} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(AP\) and \(BP\) of the string are taut. The system is in equilibrium with angle \(BAP = \alpha\) and angle \(ABP = \beta\) (see diagram). The weight of \(A\) is \(2\) N and the tensions in the parts \(AP\) and \(BP\) of the string are \(7\) N and \(T\) N respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.

1. Find the coefficient of friction between the wire and the ring \(A\). [7]
2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\). [2]
3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). [3]

\includegraphics{figure_4} A block of mass \(2\) kg is at rest on a rough horizontal plane, acted on by a force of magnitude \(12\) N at an angle of \(15°\) upwards from the horizontal (see diagram).

1. Find the frictional component of the contact force exerted on the block by the plane. [2]
2. Show that the normal component of the contact force exerted on the block by the plane has magnitude \(16.5\) N, correct to 3 significant figures. [2]

It is given that the block is on the point of sliding.

1. Find the coefficient of friction between the block and the plane. [2]

The force of magnitude \(12\) N is now replaced by a horizontal force of magnitude \(20\) N. The block starts to move.

1. Find the acceleration of the block. [5]

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]

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]

Three forces act on a particle. The first force has magnitude \(P\text{ N}\) and acts horizontally due east. The second force has magnitude \(5\text{ N}\) and acts horizontally due west. The third force has magnitude \(2P\text{ N}\) and acts vertically upwards. The resultant of these three forces has magnitude \(25\text{ N}\).

1. Calculate \(P\) and the angle between the resultant and the vertical. [7]

The particle has mass \(3\text{ kg}\) and rests on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.15\).

1. Find the acceleration of the particle, and state the direction in which it moves. [5]