Energy method - smooth inclined plane (no resistance)

1 A particle of mass 0.6 kg is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. Use an energy method to find the speed of the particle after it has moved 15 m down the plane.

1 A particle of mass 1.6 kg is projected with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a smooth plane inclined at \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.

4 A girl on a sledge starts, with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at the top of a slope of length 100 m which is at an angle of \(20 ^ { \circ }\) to the horizontal. The sledge slides directly down the slope.

1. Given that there is no resistance to the sledge's motion, find the speed of the sledge at the bottom of the slope.
2. It is given instead that the sledge experiences a resistance to motion such that the total work done against the resistance is 8500 J , and the speed of the sledge at the bottom of the slope is \(21 \mathrm {~ms} ^ { - 1 }\). Find the total mass of the girl and the sledge.

2 A particle is placed on a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The particle, of mass 4 kg , is released from rest at a point \(A\) and travels down the plane, passing through a point \(B\). The distance \(A B\) is 5 m . \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-04_371_693_500_680}

1. Find the potential energy lost as the particle moves from point \(A\) to point \(B\).
2. Hence write down the kinetic energy of the particle when it reaches point \(B\).
3. Hence find the speed of the particle when it reaches point \(B\).

4. A small block of wood, of mass 0.5 kg , slides down a line of \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-1_219_501_2042_338}
greatest slope of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\). The block is given an initial impulse of magnitude 2 Ns , and reaches the bottom of the plane with kinetic energy 19 J.

1. Find, in J , the change in the potential energy of the block as it moves down the plane.
2. Hence find the distance travelled by the block down the plane.
3. State two modelling assumptions that you have made. \section*{MECHANICS 2 (A) TEST PAPER 6 Page 2}

2 A slide at a water park may be modelled as a smooth plane of length 20 metres inclined at \(30 ^ { \circ }\) to the vertical. Anne, who has a mass of 55 kg , slides down the slide. At the top of the slide, she has an initial velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the slide.

1. Calculate Anne's initial kinetic energy.
2. By using conservation of energy, find the kinetic energy and the speed of Anne after she has travelled the 20 metres.
3. State one modelling assumption which you have made.

\includegraphics{figure_2} A particle of mass \(7.5\) kg, starting from rest at \(A\), slides down an inclined plane \(AB\). The point \(B\) is \(12.5\) metres vertically below the level of \(A\), as shown in the diagram.

1. Given that the plane is smooth, use an energy method to find the speed of the particle at \(B\). [2]
2. It is given instead that the plane is rough and the particle reaches \(B\) with a speed of \(8 \text{ ms}^{-1}\). The plane is \(25\) m long and the constant frictional force has magnitude \(F\) N. Find the value of \(F\). [3]

\(P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(PQ\).

1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\). [3]
2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J. Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\). [4]

\(P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of \(0.45 \text{ m}\) above the level of \(P\). A particle of mass \(0.3 \text{ kg}\) moves upwards along the line \(PQ\).

1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\). [3]
2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is \(0.39 \text{ J}\). Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\). [4]

A particle of mass \(0.4\) kg is projected with a speed of \(12\) m s\(^{-1}\) up a line of greatest slope of a smooth plane inclined at \(30°\) to the horizontal.

1. Find the initial kinetic energy of the particle. [1]
2. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest. [3]