Edexcel M3 (Mechanics 3) 2012 June

A particle \(P\) is moving along the positive \(x\)-axis. At time \(t = 0\), \(P\) is at the origin \(O\). At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and has velocity \(v = 2e^{-t}\) m s\(^{-1}\) in the direction of \(x\) increasing.

1. Find the acceleration of \(P\) in terms of \(x\). [3]
2. Find \(x\) in terms of \(t\). [6]

A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\). The period of the motion is \(\frac{\pi}{2}\) seconds. At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(P\) is at \(O\) and \(v = 6\). Find

1. the greatest distance of \(P\) from \(O\) during the motion, [3]
2. the greatest magnitude of the acceleration of \(P\) during the motion, [2]
3. the smallest positive value of \(t\) for which \(P\) is 1 m from \(O\). [3]

\includegraphics{figure_1} A particle \(Q\) of mass 5 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a vertical pole. Each string has length 0.6 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 1. Both strings are taut and \(Q\) is moving in a horizontal circle with constant angular speed 10 rad s\(^{-1}\). Find the tension in

1. \(AQ\),
2. \(BQ\). [10]

\includegraphics{figure_2} Figure 2 shows the cross-section \(AVBC\) of the solid \(S\) formed when a uniform right circular cone of base radius \(a\) and height \(a\), is removed from a uniform right circular cone of base radius \(a\) and height \(2a\). Both cones have the same axis \(VCO\), where \(O\) is the centre of the base of each cone.

1. Show that the distance of the centre of mass of \(S\) from the vertex \(V\) is \(\frac{5}{4}a\). [5]

The mass of \(S\) is \(M\). A particle of mass \(kM\) is attached to \(S\) at \(B\). The system is suspended by a string attached to the vertex \(V\), and hangs freely in equilibrium. Given that \(VA\) is at an angle \(45°\) to the vertical through \(V\),

1. find the value of \(k\). [5]

A fixed smooth sphere has centre \(O\) and radius \(a\). A particle \(P\) is placed on the surface of the sphere at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is released from rest at \(A\). When \(OP\) makes an angle \(\theta\) to the upward vertical through \(O\), \(P\) is on the surface of the sphere and the speed of \(P\) is \(v\). Given that \(\cos \alpha = \frac{3}{5}\)

1. show that $$v^2 = \frac{2ga}{5}(3 - 5\cos \theta)$$ [4]
2. find the speed of \(P\) at the instant when it loses contact with the sphere. [8]

\includegraphics{figure_3} Figure 3 shows a uniform equilateral triangular lamina \(PRT\) with sides of length \(2a\).

1. Using calculus, prove that the centre of mass of \(PRT\) is at a distance \(\frac{2\sqrt{3}}{3}a\) from \(R\). [6]

\includegraphics{figure_4} The circular sector \(PQU\), of radius \(a\) and centre \(P\), and the circular sector \(TUS\), of radius \(a\) and centre \(T\), are removed from \(PRT\) to form the uniform lamina \(QRSU\) shown in Figure 4.

1. Show that the distance of the centre of mass of \(QRSU\) from \(U\) is \(\frac{2a}{3\sqrt{3} - \pi}\). [6]

A particle \(B\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N. The other end of the string is attached to a fixed point \(A\). The particle is hanging in equilibrium at the point \(E\), vertically below \(A\).

1. Show that \(AE = 0.9\) m. [3]

The particle is held at \(A\) and released from rest. The particle first comes to instantaneous rest at the point \(C\).

1. Find the distance \(AC\). [5]
2. Show that while the string is taut, \(B\) is moving with simple harmonic motion with centre \(E\). [4]
3. Calculate the maximum speed of \(B\). [2]