3.03v Motion on rough surface: including inclined planes

\includegraphics{figure_7} Three points \(A\), \(B\) and \(C\) lie on a line of greatest slope of a plane inclined at an angle of \(30°\) to the horizontal, with \(AB = 1\) m and \(BC = 1\) m, as shown in the diagram. A particle of mass 0.2 kg is released from rest at \(A\) and slides down the plane. The part of the plane from \(A\) to \(B\) is smooth. The part of the plane from \(B\) to \(C\) is rough, with coefficient of friction \(\mu\) between the plane and the particle.

1. Given that \(\mu = \frac{1}{2}\sqrt{3}\), find the speed of the particle at \(C\). [8]
2. Given instead that the particle comes to rest at \(C\), find the exact value of \(\mu\). [4]

\includegraphics{figure_4} A block of mass 8 kg is placed on a rough plane which is inclined at an angle of 18° to the horizontal. The block is pulled up the plane by a light string that makes an angle of 26° above a line of greatest slope. The tension in the string is \(T\) N (see diagram). The coefficient of friction between the block and plane is 0.65.

1. The acceleration of the block is 0.2 m s\(^{-2}\). Find \(T\). [7]
2. The block is initially at rest. Find the distance travelled by the block during the fourth second of motion. [2]

A particle \(P\) of mass 0.4 kg is in limiting equilibrium on a plane inclined at \(30°\) to the horizontal.

1. Show that the coefficient of friction between the particle and the plane is \(\frac{1}{3}\sqrt{3}\). [3]

A force of magnitude 7.2 N is now applied to \(P\) directly up a line of greatest slope of the plane.

1. Given that \(P\) starts from rest, find the time that it takes for \(P\) to move 1 m up the plane. [4]

A particle \(P\) of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to \(P\).

1. Given that \(P\) is in limiting equilibrium, find the coefficient of friction between \(P\) and the plane. [3]
2. Given instead that the coefficient of friction between \(P\) and the plane is 0.3, find the distance travelled by \(P\) in the third second of its motion. [4]

\includegraphics{figure_5} A particle of mass 12 kg is going to be pulled across a rough horizontal plane by a light inextensible string. The string is at an angle of 30° above the plane and has tension \(T\) N (see diagram). The coefficient of friction between the particle and the plane is 0.5.

1. Given that the particle is on the point of moving, find the value of \(T\). [5]
2. Given instead that the particle is accelerating at 0.2 ms\(^{-2}\), find the value of \(T\). [3]

\(A\) and \(B\) are points on the same line of greatest slope of a rough plane inclined at \(30°\) to the horizontal. \(A\) is higher up the plane than \(B\) and the distance \(AB\) is \(2.25 \text{ m}\). A particle \(P\), of mass \(m \text{ kg}\), is released from rest at \(A\) and reaches \(B\) \(1.5 \text{ s}\) later. Find the coefficient of friction between \(P\) and the plane. [6]

A small box of mass 5 kg is pulled at a constant speed of \(2.5 \text{ m s}^{-1}\) down a line of greatest slope of a rough plane inclined at \(10°\) to the horizontal. The pulling force has magnitude 20 N and acts downwards parallel to a line of greatest slope of the plane.

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

The pulling force is removed while the box is moving at \(2.5 \text{ m s}^{-1}\).

1. Find the distance moved by the box after the instant at which the pulling force is removed. [4]

\includegraphics{figure_5} A particle of mass \(0.12\) kg is placed on a plane which is inclined at an angle of \(40°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting up the plane at an angle of \(30°\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.32\). Find the set of possible values of \(P\). [8]

\includegraphics{figure_6} The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). A particle \(A\) of mass \(0.3\) kg rests on the horizontal surface and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the block. The other end of the string is attached to a particle \(B\) of mass \(1.5\) kg which rests on the sloping surface of the block. The system is released from rest with the string taut.

1. Given that the block is smooth, find the acceleration of particle \(A\) and the tension in the string. [5]
2. It is given instead that the block is rough. The coefficient of friction between \(A\) and the block is \(\mu\) and the coefficient of friction between \(B\) and the block is also \(\mu\). In the first \(3\) seconds of the motion, \(A\) does not reach \(P\) and \(B\) does not reach the bottom of the sloping surface. The speed of the particles after \(3\) s is \(5\) m s\(^{-1}\). Find the acceleration of particle \(A\) and the value of \(\mu\). [9]

\includegraphics{figure_7} A rough inclined plane of length 65 cm is fixed with one end at a height of 16 cm above the other end. Particles \(P\) and \(Q\), of masses \(0.13\) kg and \(0.11\) kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley at the top of the plane. Particle \(P\) is held at rest on the plane and particle \(Q\) hangs vertically below the pulley (see diagram). The system is released from rest and \(P\) starts to move up the plane.

1. Draw a diagram showing the forces acting on \(P\) during its motion up the plane. [1]
2. Show that \(T - F > 0.32\), where \(T\) N is the tension in the string and \(F\) N is the magnitude of the frictional force on \(P\). [4]

The coefficient of friction between \(P\) and the plane is 0.6.

1. Find the acceleration of \(P\). [6]

A particle of mass 0.2 kg is resting in equilibrium on a rough plane inclined at \(20°\) to the horizontal.

1. Show that the friction force acting on the particle is 0.684 N, correct to 3 significant figures. [1]

The coefficient of friction between the particle and the plane is 0.6. A force of magnitude 0.9 N is applied to the particle down a line of greatest slope of the plane. The particle accelerates down the plane.

1. Find this acceleration. [4]

\includegraphics{figure_6} Two particles \(P\) and \(Q\), each of mass \(m\) kg, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The plane is inclined at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). Particle \(P\) rests on the plane and particle \(Q\) hangs vertically, as shown in the diagram. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium.

1. Show that the coefficient of friction between \(P\) and the plane is \(\frac{4}{3}\). [5]

A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at 2.5 m s\(^{-2}\).

1. Find the value of \(m\). [5]

A block of mass \(5\) kg is being pulled by a rope up a rough plane inclined at \(6°\) to the horizontal. The rope is parallel to a line of greatest slope of the plane and the block is moving at constant speed. The coefficient of friction between the block and the plane is \(0.3\). Find the tension in the rope. [4]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of masses \(0.4\) kg and \(0.7\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The coefficient of friction between \(P\) and the plane is \(0.5\). The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) lies on the plane and particle \(Q\) hangs vertically. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). A force of magnitude \(X\) N, acting directly down the plane, is applied to \(P\).

1. Show that the greatest value of \(X\) for which \(P\) remains stationary is \(6.2\). [4]
2. Given instead that \(X = 0.8\), find the acceleration of \(P\). [4]

A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.

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

The force of magnitude 15 N is now replaced by a force of magnitude \(X\) N acting up the line of greatest slope.

1. Find the greatest value of \(X\) for which the block does not move. [2]

A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac{24}{25}\). The force is applied for a period of 5 s, during which time the block moves a distance of 4.5 m.

1. Find the magnitude of the frictional force on the block. [4]
2. Show that the coefficient of friction between the block and the plane is 0.165, correct to 3 significant figures. [3]
3. When the block has moved a distance of 4.5 m, the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion. [4]

\(O\) and \(A\) are fixed points on a rough horizontal surface, with \(OA = 1 \text{ m}\). A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally with speed \(U \text{ m s}^{-1}\) from \(A\) in the direction \(OA\) and moves in a straight line. After projection, when the displacement of \(P\) from \(O\) is \(x \text{ m}\), the velocity of \(P\) is \(v \text{ m s}^{-1}\). The coefficient of friction between the surface and \(P\) is \(0.4\). A force of magnitude \(\frac{0.8}{x} \text{ N}\) acts on \(P\) in the direction \(PO\).

1. Show that, while the particle is in motion, \(v \frac{\text{d}v}{\text{d}x} = -4 - \frac{2}{x}\). [3]

It is given that \(P\) comes to instantaneous rest between \(x = 2.0\) and \(x = 2.1\).

1. Find the set of possible values of \(U\). [5]

\includegraphics{figure_6} A particle \(P\) of mass \(0.2\) kg is projected with velocity \(2\) m s\(^{-1}\) upwards along a line of greatest slope on a plane inclined at \(30°\) to the horizontal (see diagram). Air resistance of magnitude \(0.5v\) N opposes the motion of \(P\), where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s after projection. The coefficient of friction between \(P\) and the plane is \(\frac{1}{2\sqrt{3}}\). The particle \(P\) reaches a position of instantaneous rest when \(t = T\).

1. Show that, while \(P\) is moving up the plane, \(\frac{dv}{dt} = -2.5(3 + v)\). [3]
2. Calculate \(T\). [4]
3. Calculate the speed of \(P\) when \(t = 2T\). [5]

At the point \(C\) the horizontal surface becomes rough, with coefficient of friction \(\mu\) between the combined particle and the surface. The deceleration of the combined particle at \(C\) is \(\frac{g}{20}\).

1. Find the value of \(\mu\). [4]

A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\). \includegraphics{figure_4}

1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
2. Find the value of \(\mu\). [3]

Two particles \(A\) and \(B\) of masses \(m\) and \(km\) respectively are connected by a light inextensible string of length \(a\). The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle \(A\) is at a distance \(a\) from the centre of the turntable and particle \(B\) is at a distance \(2a\) from the centre of the turntable. The coefficient of friction between each particle and the turntable is \(\frac{1}{3}\). When the turntable is made to rotate with angular speed \(\frac{2}{5}\sqrt{\frac{g}{a}}\), the system is in limiting equilibrium.

1. Find the tension in the string, in terms of \(m\) and \(g\). [4]
2. Find the value of \(k\). [3]

\includegraphics{figure_4} The end \(A\) of a uniform rod \(AB\) of length \(6a\) and weight \(W\) is in contact with a rough vertical wall. One end of a light inextensible string of length \(3a\) is attached to the midpoint \(C\) of the rod. The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The rod is in equilibrium when the angle between the rod and the wall is \(\theta\), where \(\tan \theta = \frac{3}{4}\). A particle of weight \(W\) is attached to the point \(E\) on the rod, where the distance \(AE\) is equal to \(ka\) (\(3 < k < 6\)) (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\frac{1}{3}\). The rod is about to slip down the wall.

1. Find the value of \(k\). [5]
2. Find, in terms of \(W\), the magnitude of the frictional force between the rod and the wall. [2]

The tile on a roof becomes loose and slides from rest down the roof. The roof is modelled as a plane surface inclined at 30° to the horizontal. The coefficient of friction between the tile and the roof is 0.4. The tile is modelled as a particle of mass \(m\) kg.

1. Find the acceleration of the tile as it slides down the roof. [7]

The tile moves a distance 3 m before reaching the edge of the roof.

1. Find the speed of the tile as it reaches the edge of the roof. [2]
2. Write down the answer to part (a) if the tile had mass \(2m\) kg. [1]

\includegraphics{figure_3} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at 20° to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4. The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate

1. the normal reaction of the plane on \(P\), [2]
2. the value of \(X\). [4]

The force of magnitude \(X\) newtons is now removed.

1. Show that \(P\) remains in equilibrium on the plane. [4]

\includegraphics{figure_2} A parcel of weight \(10\) N lies on a rough plane inclined at an angle of \(30°\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is \(18\) N. The coefficient of friction between the parcel and the plane is \(\mu\). Find

1. the value of \(P\), [4]
2. the value of \(\mu\). [5]

The horizontal force is removed.

1. Determine whether or not the parcel moves. [5]