3.03r Friction: concept and vector form

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]

Lizzie is sat securely on a wooden sledge. The combined mass of Lizzie and the sledge is \(M\) kilograms. The sledge is being pulled forward in a straight line along a horizontal surface by means of a light inextensible rope, which is attached to the front of the sledge. This rope stays inclined at an acute angle \(\theta\) above the horizontal and remains taut as the sledge moves forward. \includegraphics{figure_17} The sledge remains in contact with the surface throughout. The coefficient of friction between the sledge and the surface is \(\mu\) and there are no other resistance forces. Lizzie and the sledge move forward with constant acceleration, \(a \text{ m s}^{-2}\) The tension in the rope is a constant \(T\) Newtons.

1. Show that $$T = \frac{M(a + \mu g)}{\cos \theta + \mu \sin \theta}$$ [7 marks]
2. It is known that when \(M = 30\), \(\theta = 30°\), and \(T = 40\), the sledge remains at rest. Lizzie uses these values with the relationship formed in part (a) to find the value for \(\mu\) Explain why her value for \(\mu\) may be incorrect. [2 marks]

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]

A particle P, of mass \(m\), moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude \(\frac{kmg}{x^2}\), where \(x\) is the distance OP. The coefficient of friction between P and the table is \(\mu\). P is initially projected in a direction directly away from O. The velocity of P is first zero at a point A which is a distance \(a\) from O.

1. Show that the velocity \(v\) of P, when P is moving away from O, satisfies the differential equation $$\frac{\mathrm{d}}{\mathrm{d}x}(v^2) + \frac{2kg}{x^2} + 2\mu g = 0.$$ [3]
2. Verify that $$v^2 = 2gk\left(\frac{1}{x} - \frac{1}{a}\right) + 2\mu g(a-x).$$ [3]
3. Find, in terms of \(k\) and \(a\), the range of values of \(\mu\) for which P remains at A. [2]

A particle, of mass 2 kg, is placed at a point A on a rough horizontal surface. There is a straight vertical wall on the surface and the point on the wall nearest to A is B. The distance AB is 5 m. The particle is projected with speed 4.2 m s\(^{-1}\) along the surface from A towards B. The particle hits the wall directly and rebounds. The coefficient of friction between the particle and the surface is 0.1.

1. Determine the speed of the particle immediately before impact with the wall. [4]

The magnitude of the impulse that the wall exerts on the particle is 9.8 N s.

1. Find the speed of the particle immediately after impact with the wall. [2]

In this question the box should be modelled as a particle. A box of mass \(m\) kg is placed on a rough slope which makes an angle of \(\alpha\) with the horizontal.

1. Show that the box is on the point of slipping if \(\mu = \tan \alpha\), where \(\mu\) is the coefficient of friction between the box and the slope. [2]

A box of mass 5 kg is pulled up a rough slope which makes an angle of 15° with the horizontal. The box is subject to a constant frictional force of magnitude 3 N. The speed of the box increases from 2 m s\(^{-1}\) at a point A on the slope to 5 m s\(^{-1}\) at a point B on the slope with B higher up the slope than A. The distance AB is 10 m. \includegraphics{figure_6} The pulling force has constant magnitude \(P\) N and acts at a constant angle of 25° above the slope, as shown in the diagram.

1. Use the work–energy principle to determine the value of \(P\). [5]

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_11} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC, with AB perpendicular to AC, which lies in a vertical plane. The length of AB is 3 cm, and the length of AC is 12 cm. The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle \(\theta\) with the horizontal. A horizontal force of magnitude \(P\) N is now applied to the prism at B. This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each \(\frac{1}{2}\).

1. Show that the least value of \(P\) needed to move the prism is given by $$P = \frac{40 \cos \theta + 95 \sin \theta}{16 \sin \theta - 13 \cos \theta}.$$ [8]
2. Determine the range in which the value of \(\theta\) must lie. [4]

\includegraphics{figure_4} A uniform rod AB has mass 3 kg and length 4 m. The end A of the rod is in contact with rough horizontal ground. The rod rests in equilibrium on a smooth horizontal peg 1.5 m above the ground, such that the rod is inclined at an angle of \(25°\) to the ground (see diagram). The rod is in a vertical plane perpendicular to the peg.

1. Determine the magnitude of the normal contact force between the peg and the rod. [3]
2. Determine the range of possible values of the coefficient of friction between the rod and the ground. [5]

\includegraphics{figure_14} One end of a light inextensible string is attached to a particle \(A\) of mass \(2\text{kg}\). The other end of the string is attached to a second particle \(B\) of mass \(3\text{kg}\). 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 \(4\text{kg}\). 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 \(a\text{ms}^{-2}\).

1. By considering an equation involving \(\mu\), \(a\) and \(g\) show that \(a < \frac{5}{9}g(2\sqrt{3} - 1)\). [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]

A body of mass 20 kg is on a rough plane inclined at angle \(\alpha\) to the horizontal. The body is held at rest on the plane by the action of a force of magnitude \(P\) N. The force is acting up the plane in a direction parallel to a line of greatest slope of the plane. The coefficient of friction between the body and the plane is \(\mu\).

1. When \(P = 100\), the body is on the point of sliding down the plane. Show that \(g \sin \alpha = g\mu \cos \alpha + 5\). [4]
2. When \(P\) is increased to 150, the body is on the point of sliding up the plane. Use this, and your answer to part (a), to find an expression for \(\alpha\) in terms of \(g\). [3]

\includegraphics{figure_13} Particles \(A\) and \(B\) of masses \(2m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45°\) and \(B\) is above the plane. The vertical plane defined by \(APB\) contains a line of greatest slope of the plane, and \(PA\) is inclined at angle \(2\alpha\) to the horizontal (see diagram).

1. Show that the normal reaction \(R\) between \(A\) and the plane is \(mg(2 \cos \alpha - \sin \alpha)\). [3]
2. Show that \(R \geqslant \frac{1}{2}mg\sqrt{2}\). [3]

The coefficient of friction between \(A\) and the plane is \(\mu\). The particle \(A\) is about to slip down the plane.

1. Show that \(0.5 < \tan \alpha \leqslant 1\). [3]
2. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies. [3]

A particle of mass \(m\) is placed on a rough inclined plane. The plane makes an angle \(\theta\) with the horizontal. The coefficient of friction between the particle and the plane is \(\mu\) where \(\mu < \tan \theta\). The particle is released from rest and accelerates down the plane.

1. Draw a fully labelled diagram to show the forces acting on the particle. [1]
2. Find an expression in terms of \(g\), \(\theta\) and \(\mu\) for the acceleration of the particle. [5]
3. Explain what would happen to the particle if \(\mu > \tan \theta\). [1]

A particle \(P\) of mass \(2\) kg rests on a long rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.2\). A light inextensible string has one end attached to \(P\) and the other end attached to a particle \(Q\) of mass \(4\) kg. The particle \(Q\) is placed on a smooth plane inclined at \(30^{\circ}\) to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time \(t = 0\), where \(t\) is measured in seconds. In the subsequent motion \(P\) does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(\frac{8}{3}\) m s\(^{-2}\). [4]

After the particles have moved a distance of \(12\) m the string is cut.

1. Find the corresponding value of \(t\) and the speed of the particles at this instant. [4]
2. Find the value of \(t\) when \(P\) comes to rest. [3]