Energy methods on slope

7 A box of mass 50 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal.

1. Find an inequality for the coefficient of friction between the box and the plane. In fact the coefficient of friction between the box and the plane is 0.19 .
2. A girl pushes the box with a force of 50 N , acting down a line of greatest slope of the plane, for a distance of 5 m . She then stops pushing. Use an energy method to find the speed of the box when it has travelled a further 5 m . The box then comes to a plane inclined at \(20 ^ { \circ }\) below the horizontal. The box moves down a line of greatest slope of this plane. The coefficient of friction is still 0.19 and the girl is not pushing the box.
3. Find the acceleration of the box.

1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A particle \(P\) of mass \(m\) is held at rest at a point \(A\) on the plane.
The particle is then projected with speed \(u\) up a line of greatest slope of the plane and comes to instantaneous rest at the point \(B\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 7 }\)

1. Show that the magnitude of the frictional force acting on the particle, as it moves from \(A\) to \(B\), is \(\frac { 4 m g } { 35 }\) Given that \(u = \sqrt { 10 a g }\), use the work-energy principle
2. to find \(A B\) in terms of \(a\),
3. to find, in terms of \(a\) and \(g\), the speed of \(P\) when it returns to \(A\).

1. \begin{figure}[h] \end{figure} A small book of mass \(m\) is held on a rough straight desk lid which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The book is released from rest at a distance of 0.5 m from the edge of the desk lid, as shown in Figure 1. The book slides down the desk lid and then hits the floor that is 0.8 m below the edge of the desk lid. The coefficient of friction between the book and the desk lid is 0.4 The book is modelled as a particle which, after leaving the desk lid, is assumed to move freely under gravity.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction on the book as it slides down the desk lid.
2. Use the work-energy principle to find the speed of the book as it hits the floor.

3. \begin{figure}[h] \end{figure} Figure 1 shows part of the end elevation of a building which sits on horizontal ground. The side of the building is vertical and has height \(h\). A small stone of mass \(m\) is at rest on the roof of the building at the point \(A\). The stone slides from rest down a line of greatest slope of the roof and reaches the edge \(B\) of the roof with speed \(\sqrt { 2 g h }\) The stone then moves under gravity before hitting the ground with speed \(W\).
In a model of the motion of the stone from \(\boldsymbol { B }\) to the ground

• the stone is modelled as a particle
• air resistance is ignored

Using the principle of conservation of mechanical energy and the model,

1. find \(W\) in terms of \(g\) and \(h\). In a model of the motion of the stone from \(\boldsymbol { A }\) to \(\boldsymbol { B }\)
   Using this model,
2. find, in terms of \(m\) and \(g\), the magnitude of the frictional force acting on the stone as it slides down the roof,
3. use the work-energy principle to find \(d\) in terms of \(h\).

\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]

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_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]