3.03v Motion on rough surface: including inclined planes

\includegraphics{figure_10} \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30°\) to the horizontal. The distance \(AB\) is \(20\text{m}\). \(M\) is a point on the plane between \(A\) and \(B\). The surface of the plane is smooth between \(A\) and \(M\), and rough between \(M\) and \(B\). A particle \(P\) is projected with speed \(4.2\text{m s}^{-1}\) from \(A\) down the line of greatest slope (see diagram). \(P\) moves down the plane and reaches \(B\) with speed \(12.6\text{m s}^{-1}\). The coefficient of friction between \(P\) and the rough part of the plane is \(\frac{\sqrt{3}}{6}\).

1. Find the distance \(AM\). [8]
2. Find the angle between the contact force and the downward direction of the line of greatest slope when \(P\) is in motion between \(M\) and \(B\). [3]

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]

Particle \(A\), of mass \(m\) kg, lies on the plane \(\Pi_1\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. Particle \(B\), of \(4m\) kg, lies on the plane \(\Pi_2\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\). The coefficient of friction between particle \(A\) and \(\Pi_1\) is \(\frac{1}{4}\) and plane \(\Pi_2\) is smooth. Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below. \includegraphics{figure_13}

1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac{7g}{15}\) m s\(^{-2}\). [6]
2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac{1}{4}\) m when its speed is \(\sqrt{\frac{7g}{30}}\) m s\(^{-1}\). [2]

A particle is projected from a point \(P\) on an inclined plane, up the line of greatest slope through \(P\), with initial speed \(V\). The angle of the plane to the horizontal is \(\theta\).

1. If the plane is smooth, and the particle travels for a time \(\frac{2V}{g}\cos\theta\) before coming instantaneously to rest, show that \(\theta = \frac{1}{4}\pi\). [4]
2. If the same plane is given a roughened surface, with a coefficient of friction 0.5, find the distance travelled before the particle comes instantaneously to rest. [5]

\includegraphics{figure_12} A particle \(P\) of mass 2 kg can move along a line of greatest slope on a smooth plane, inclined at \(30°\) to the horizontal. \(P\) is initially at rest at a point on the plane, and a force of constant magnitude 20 N is applied to \(P\) parallel to and up the slope (see diagram).

1. Copy and complete the diagram, showing all forces acting on \(P\). [1]
2. Find the velocity of \(P\) in terms of time \(t\) seconds, whilst the force of 20 N is applied. [4]

After 3 seconds the force is removed at the instant that \(P\) collides with a particle of mass 1 kg moving down the slope with speed 5 m s\(^{-1}\). The coefficient of restitution between the particles is 0.2.

1. Express the velocity of \(P\) as a function of time after the collision. [6]

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 stone that weighs 15 kg is propelled across the ice in an ice rink with an initial speed of \(4 \text{ m s}^{-1}\). The coefficient of friction between the stone and the ice is \(0.017\). How far does the stone slide before it comes to rest? [5]

\includegraphics{figure_11} The diagram shows a particle, \(A\), of mass \(m_1\) at rest on a rough slope at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). Particle \(A\) is connected by a light inextensible string to another particle, \(B\), of mass \(m_2\). The string passes over a smooth peg at the top of the slope and particle \(B\) is hanging freely.

1. In the case when \(m_2 = \frac{1}{4}m_1\), particle \(A\) is on the point of sliding down the slope.
   1. Draw a fully labelled diagram to show all the forces acting on the particles. [2]
   2. Find the coefficient of friction between \(A\) and the slope. [6]
2. In the case when \(m_2 = m_1\), find the acceleration of the particles. [4]

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]