Edexcel M3 (Mechanics 3) 2002 January

A particle \(P\) of mass 0.2 kg moves away from the origin along the positive \(x\)-axis. It moves under the action of a force directed away from the origin \(O\), of magnitude \(\frac{5}{x+1}\) N, where \(OP = x\) metres. Given that the speed of \(P\) is 5 m s\(^{-1}\) when \(x = 0\), find the value of \(x\), to 3 significant figures, when the speed of \(P\) is 15 m s\(^{-1}\). [8]

One end of a light elastic string, of natural length 2 m and modulus of elasticity 19.6 N, is attached to a fixed point \(A\). A small ball \(B\) of mass 0.5 kg is attached to the other end of the string. The ball is released from rest at \(A\) and first comes to instantaneous rest at the point \(C\), vertically below \(A\).

1. Find the distance \(AC\). [6]
2. Find the instantaneous acceleration of \(B\) at \(C\). [3]

\includegraphics{figure_1} A rod \(AB\), of mass \(2m\) and length \(2a\), is suspended from a fixed point \(C\) by two light strings \(AC\) and \(BC\). The rod rests horizontally in equilibrium with \(AC\) making an angle \(\alpha\) with the rod, where \(\tan \alpha = \frac{3}{4}\), and with \(AC\) perpendicular to \(BC\), as shown in Fig. 1.

1. Give a reason why the rod cannot be uniform. [1]
2. Show that the tension in \(BC\) is \(\frac{4}{5}mg\) and find the tension in \(AC\). [5]

The string \(BC\) is elastic, with natural length \(a\) and modulus of elasticity \(kmg\), where \(k\) is constant.

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

\includegraphics{figure_2} Figure 2 shows the region \(R\) bounded by the curve with equation \(y^2 = rx\), where \(r\) is a positive constant, the \(x\)-axis and the line \(x = r\). A uniform solid of revolution \(S\) is formed by rotating \(R\) through one complete revolution about the \(x\)-axis.

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac{4}{5}r\). [6]

The solid is placed with its plane face on a plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. Given that \(S\) does not topple,

1. find, to the nearest degree, the maximum value of \(\alpha\). [4]

A cyclist is travelling around a circular track which is banked at 25° to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6. The cyclist moves with constant speed in a horizontal circle of radius 40 m, without the tyres slipping. Find the maximum speed of the cyclist. [10]

The points \(O\), \(A\), \(B\) and \(C\) lie in a straight line, in that order, where \(OA = 0.6\) m, \(OB = 0.8\) m and \(OC = 1.2\) m. A particle \(P\), moving along this straight line, has a speed of \(\left(\frac{1}{10}\sqrt{5}\right)\) m s\(^{-1}\) at \(A\), \(\left(\frac{1}{5}\sqrt{5}\right)\) m s\(^{-1}\) at \(B\) and is instantaneously at rest at \(C\).

1. Show that this information is consistent with \(P\) performing simple harmonic motion with centre \(O\). [5]

Given that \(P\) is performing simple harmonic motion with centre \(O\),

1. show that the speed of \(P\) at \(O\) is 0.6 m s\(^{-1}\), [2]
2. find the magnitude of the acceleration of \(P\) as it passes \(A\), [2]
3. find, to 3 significant figures, the time taken for \(P\) to move directly from \(A\) to \(B\). [4]

\includegraphics{figure_3} Figure 3 shows a fixed hollow sphere of internal radius \(a\) and centre \(O\). A particle \(P\) of mass \(m\) is projected horizontally from the lowest point \(A\) of a sphere with speed \(\sqrt{\left(\frac{5}{4}ag\right)}\). It moves in a vertical circle, centre \(O\), on the smooth inner surface of the sphere. The particle passes through the point \(B\), which is in the same horizontal plane as \(O\). It leaves the surface of the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical.

1. Find, in terms of \(m\) and \(g\), the normal reaction between \(P\) and the surface of the sphere at \(B\). [4]
2. Show that \(\theta = 60°\). [7]

After leaving the surface of the sphere, \(P\) meets it again at the point \(A\).

1. Find, in terms of \(a\) and \(g\), the time \(P\) takes to travel from \(C\) to \(A\). [4]