6.02c Work by variable force: using integration

8 A car of mass 1 tonne reaches the foot of an incline travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It reaches the top of the incline 50 seconds later travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The length of the incline is 1200 m and the angle made with the horizontal is \(\sin ^ { - 1 } \left( \frac { 1 } { 8 } \right)\). The constant resistance to motion is 400 N . Find the average power developed by the engine of the car.

A particle \(P\) of mass 0.5 kg moves along a straight line. When \(P\) is at a distance \(x\) m from a fixed point \(O\) on the line, the force acting on it is directed towards \(O\) and has magnitude \(\frac{8}{x}\) N. When \(x = 2\), the speed of \(P\) is 4 ms\(^{-1}\). Find the speed of \(P\) when it is 0.5 m from \(O\). [8 marks]

A particle \(P\) of mass \(m\) kg moves vertically upwards under gravity, starting from ground level. It is acted on by a resistive force of magnitude \(m f(x)\) N, where \(f(x)\) is a function of the height \(x\) m of \(P\) above the ground. When \(P\) is at this height, its upward speed \(v\) ms\(^{-1}\) is given by $$v^2 = 2e^{-2gx} - 1.$$

1. Write down a differential equation for the motion of \(P\) and hence determine \(f(x)\) in terms of \(g\) and \(x\). [5 marks]
2. Show that the greatest height reached by \(P\) above the ground is \(\frac{1}{2g} \ln 2\) m. [2 marks]

Given that the work, in J, done by \(P\) against the resisting force as it moves from ground level to a point \(H\) m above the ground is equal to \(\int_0^H m f(x) dx\),

1. show that the total work done by \(P\) against the resistance during its upward motion is \(\frac{1}{2}m(1 - \ln 2)\) J. [3 marks]

In this question use \(g = 9.8 \text{ m s}^{-2}\) A lift is used to raise a crate of mass 250 kg The lift exerts an upward force of magnitude \(P\) newtons on the crate. When the crate is at a height of \(x\) metres above its initial position $$P = k(x + 1)(12 - x) + 2450$$ where \(k\) is a constant. The crate is initially at rest, at the point where \(x = 0\)

1. Show that the work done by the upward force as the crate rises to a height of 12 metres is given by $$29400 + 360k$$ [3 marks]
2. The speed of the crate is \(3 \text{ m s}^{-1}\) when it has risen to a height of 12 metres. Find the speed of the crate when it has risen to a height of 15 metres. [5 marks]
3. Find the height of the crate when its speed becomes zero. [2 marks]
4. Air resistance has been ignored. Explain why this is reasonable in this context. [1 mark]

A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v^2\).

1. Show that \(v^2 = \frac{1}{k}\left(T - Ae^{-\frac{2kx}{m}}\right)\) where \(A\) and \(k\) are constants. [5]

\(P\) starts from rest at \(O\).

1. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\). [4]
2. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases. [1]

7 A small ball \(B\) of mass 0.2 kg moves in a narrow fixed smooth cylindrical tube \(O A\) of length 1 m , closed at the end \(A\). When the ball has displacement \(x \mathrm {~m}\) from \(O\), it has velocity \(v \mathrm {~ms} ^ { - 1 }\) in the direction \(O A\) and experiences a resisting force of magnitude \(\frac { k } { 1 - x } \mathrm {~N}\).

1. \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-4_186_805_488_715} The tube is fixed in a horizontal position and \(B\) is projected from \(O\) towards \(A\) with velocity \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Given that \(B\) comes to instantaneous rest after travelling 0.55 m , show that \(k = 0.1803\), correct to 4 significant figures.
2. The tube is now fixed in a vertical position with \(O\) above \(A\). The ball \(B\) is released from rest at \(O\). Calculate the speed of \(B\) after it has descended 0.1 m . \end{document}