3.03u Static equilibrium: on rough surfaces

\includegraphics{figure_2} A small packet of mass 0.3 kg rests on a rough horizontal surface. The coefficient of friction between the packet and the surface is \(\frac{1}{4}\). Two strings are attached to the packet, making angles of 45° and 30° with the horizontal, and when forces of magnitude 2 N and \(F\) N are exerted through the strings as shown, the packet is on the point of moving in the direction \(\overrightarrow{AB}\). Find the value of \(F\). \hfill [7 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 block of mass \(m\) kg is at rest on a horizontal plane. The coefficient of friction between the block and the plane is \(0.2\).

1. When a horizontal force of magnitude \(5\) N acts on the block, the block is on the point of slipping. Find the value of \(m\). [3]

1. \includegraphics{figure_5ii} When a force of magnitude \(P\) N acts downwards on the block at an angle \(\alpha\) to the horizontal, as shown in the diagram, the frictional force on the block has magnitude \(6\) N and the block is again on the point of slipping. Find
   1. the value of \(\alpha\) in degrees,
   2. the value of \(P\).
   [8]

\includegraphics{figure_3} A block \(B\) of mass \(0.4\) kg and a particle \(P\) of mass \(0.3\) kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).

1. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table. [5]
2. A horizontal force of magnitude \(X\) N, acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\). [3]

\includegraphics{figure_3} A block \(B\) of mass 0.4 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).

1. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table. [5]
2. A horizontal force of magnitude \(X\) N, acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\). [3]

\includegraphics{figure_2} Figure 2 shows a large block of mass 50 kg being pulled on rough horizontal ground by means of a rope attached to the block. The tension in the rope is 200 N and it makes an angle of 40° with the horizontal. Under these conditions, the block is on the point of moving. Modelling the block as a particle,

1. show that the coefficient of friction between the block and the ground is 0.424 correct to 3 significant figures. [6 marks]

The angle with the horizontal at which the rope is being pulled is reduced to 30°. Ignoring air resistance and assuming that the tension in the rope and the coefficient of friction remain unchanged,

1. find the acceleration of the block. [6 marks]

\includegraphics{figure_2} A particle \(P\), of mass 2 kg, lies on a rough plane inclined at an angle of 30° to the horizontal. A force \(H\), whose line of action is parallel to the line of greatest slope of the plane, is applied to the particle as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{\sqrt{3}}\). Given that the particle is on the point of moving up the plane,

1. draw a diagram showing all the forces acting on the particle, [2 marks]
2. show that the ratio of the magnitude of the frictional force to the magnitude of \(H\) is equal to \(1 : 2\) [7 marks]

The force \(H\) is now removed but \(P\) remains at rest.

1. Use the principle of friction to explain how this is possible. [2 marks]

A sledge of mass 4 kg rests in limiting equilibrium on a rough slope inclined at an angle 10° to the horizontal. By modelling the sledge as a particle,

1. show that the coefficient of friction, \(\mu\), between the sledge and the ground is 0.176 correct to 3 significant figures. [6 marks]

The sledge is placed on a steeper part of the slope which is inclined at an angle 30° to the horizontal. The value of \(\mu\) remains unchanged.

1. Find the minimum extra force required along the line of greatest slope to prevent the sledge from slipping down the hill. [5 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]

A uniform ladder \(AB\), of length 6 metres and mass 22 kg, rests with its foot, \(A\), on rough horizontal ground. The ladder rests against the top of a smooth vertical wall at the point \(C\), where the length \(AC\) is 5 metres. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the ground is \(60°\). A man, of mass 88 kg, is standing on the ladder. The man may be modelled as a particle at the point \(D\), where the length of \(AD\) is 4 metres. The ladder is on the point of slipping. \includegraphics{figure_7}

1. Draw a diagram to show the forces acting on the ladder. [2 marks]
2. Find the coefficient of friction between the ladder and the horizontal ground. [6 marks]

A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2\mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.

1. Draw a diagram to show the forces acting on the ladder. [2 marks]
2. Find \(\tan \theta\) in terms of \(\mu\). [7 marks]

A uniform plank of wood \(XY\), of mass 1.4 kg, rests with its upper end \(X\) against a rough vertical wall and its lower end \(Y\) on rough horizontal ground. The coefficient of friction between the plank and both the wall and the ground is \(\mu\). The plank is in limiting equilibrium at both ends and the vertical component of the force exerted on the plank by the ground has magnitude 12 N. Find the value of \(\mu\), to 2 decimal places. [8 marks]

\includegraphics{figure_5} A uniform rod \(AB\), of mass 3 kg and length 4 m, is in limiting equilibrium with \(A\) on rough horizontal ground. The rod is at an angle of 60° to the horizontal and is supported by a small smooth peg \(P\), such that the distance \(AP\) is 2.5 m (see diagram). Find

1. the force acting on the rod at \(P\), [3]
2. the coefficient of friction between the ground and the rod. [5]

A uniform ladder \(AB\), of weight \(W\) and length \(2a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{12}{13}\). A man of weight \(6W\) is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{5W}{24}\left(1 + \frac{6x}{a}\right)\). [5]

The coefficient of friction between the ladder and the ground is \(\frac{1}{3}\).

1. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. [3]

The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight \(6W\) can now stand at the top of the ladder \(B\) without the ladder slipping.

1. Find the least possible value of \(\tan \alpha\). [3]

Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}

1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
   1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
   2. Calculate the work done against friction per tile. [2]
   3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]

A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. \includegraphics{figure_3}

1. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. [5]

The rectangles BCJA and FGHI are folded through 90° about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.

1. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). [4]

The \(x\)- and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force \(P\) N is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH.

1. Calculate the value of \(P\). [5]

The coefficient of friction between the fire-screen and the floor is \(\mu\).

1. For what values of \(\mu\) does the fire-screen slide before it tips? [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]

The diagram shows a particle of mass \(0.7\) kg resting on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.25\). A light elastic string, of natural length \(50\) cm and modulus of elasticity \(6.86\) N, is attached to the particle. The string is kept at an angle of \(60°\) to the horizontal and is gradually extended by pulling on it until the particle moves. Show that the particle starts to move when the extension in the string is \(17\) cm. \includegraphics{figure_2} [8 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85

1. The boy applies a horizontal force of 150 N. Show that the crate remains stationary. [3 marks]
2. Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N, at an angle of 15° above the horizontal, as shown in the diagram. \includegraphics{figure_5} Determine whether the crate remains stationary. Fully justify your answer. [5 marks]

A wooden crate rests on a rough horizontal surface. The coefficient of friction between the crate and the surface is 0.6 A forward force acts on the crate, parallel to the surface. When this force is 600 N, the crate is on the point of moving. Find the weight of the crate. Circle your answer. [1 mark] 1000 N \quad 100 kg \quad 360 N \quad 36 kg

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]

One end of a rope is attached to a block A of mass 2 kg. The other end of the rope is attached to a second block B of mass 4 kg. Block A is held at rest on a fixed rough ramp inclined at \(30°\) to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P, at a distance \(d\) m above the ground, as shown in the diagram. \includegraphics{figure_7} Block A is more than \(d\) m from P. The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is \(\frac{1}{2\sqrt{3}}\). The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.

1. Determine, in terms of \(g\) and \(d\), the work done against friction as A moves \(d\) m up the ramp. [3]
2. Given that the speed of B immediately before it hits the ground is \(1.75 \text{ m s}^{-1}\), use the work–energy principle to determine the value of \(d\). [5]
3. Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic. [1]

\includegraphics{figure_9} Fig. 9 shows a uniform rod AB of length \(2a\) and weight \(8W\) which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight \(W\) is threaded on the bar and is connected to the rod at B by a light inextensible string of length \(2a\). The system is in equilibrium with the rod inclined at an angle \(\theta\) to the horizontal.

1. Determine, in terms of \(W\) and \(\theta\), the tension in the string. [4] It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is \(2.4a\).
2. Determine the coefficient of friction between the bar and the ring. [6]

A uniform ladder of length 8 m and weight 180 N stands on a rough horizontal surface and rests against a smooth vertical wall. The ladder makes an angle of 20° with the wall. A woman of weight 720 N stands on the ladder. Fig. 7 shows this situation modelled with the woman's weight acting at a distance \(x\) m from the lower end of the ladder. The system is in equilibrium. \includegraphics{figure_7}

1. Show that the frictional force between the ladder and the horizontal surface is \(F\) N, where \(F = 90(1 + x)\tan 20°\). [4]
2. 1. State with a reason whether \(F\) increases, stays constant or decreases as \(x\) increases. [1]
   2. Hence determine the set of values of the coefficient of friction between the ladder and the surface for which the woman can stand anywhere on the ladder without it slipping. [4]