Edexcel M3 (Mechanics 3) 2014 June

A particle \(P\) of mass \(0.25\) kg is moving along the positive \(x\)-axis under the action of a single force. At time \(t\) seconds \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v\) m s\(^{-1}\) where \(\frac{\mathrm{d}v}{\mathrm{d}x} = 3\). It is given that \(x = 2\) and \(v = 3\) when \(t = 0\)

1. Find the magnitude of the force acting on \(P\) when \(x = 5\) [4]
2. Find the value of \(t\) when \(x = 5\) [4]

\includegraphics{figure_1} A cone of semi-vertical angle \(60°\) is fixed with its axis vertical and vertex upwards. A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60°\) with the horizontal, as shown in Figure 1.

1. Find the tension in the string, in terms of \(m\), \(l\), \(\omega\) and \(g\). [7]

The particle remains on the surface of the cone.

1. Show that the time for the particle to make one complete revolution is greater than $$2\pi\sqrt{\frac{l\sqrt{3}}{2g}}$$ [6]

One end \(A\) of a light elastic string \(AB\), of modulus of elasticity \(mg\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(AB\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(AB = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.

1. Show that when the particle comes to rest it has moved a distance \(2a(\sin \theta - \mu \cos \theta)\) down the plane. [6]
2. Given that there is no further motion, show that \(\mu \geqslant \frac{1}{3} \tan \theta\). [5]

\includegraphics{figure_2} A smooth sphere of radius \(a\) is fixed with a point \(A\) of its surface in contact with a fixed vertical wall. A particle is placed on the highest point of the sphere and is projected towards the wall and perpendicular to the wall with horizontal speed \(\sqrt{\frac{2ag}{5}}\), as shown in Figure 2. The particle leaves the surface of the sphere with speed \(V\).

1. Show that \(V = \sqrt{\frac{4ag}{5}}\) [7]

The particle strikes the wall at the point \(X\).

1. Find the distance \(AX\). [9]

\includegraphics{figure_3} A uniform solid right circular cylinder has height \(h\) and radius \(r\). The centre of one plane face is \(O\) and the centre of the other plane face is \(Y\). A cylindrical hole is made by removing a solid cylinder of radius \(\frac{1}{4}r\) and height \(\frac{1}{4}h\) from the end with centre \(O\). The axis of the cylinder removed is parallel to \(OY\) and meets the end with centre \(O\) at \(X\), where \(OX = \frac{1}{4}r\). One plane face of the cylinder removed coincides with the plane face through \(O\) of the original cylinder. The resulting solid \(S\) is shown in Figure 3.

1. Show that the centre of mass of \(S\) is at a distance \(\frac{85h}{168}\) from the plane face containing \(O\). [7]

The solid \(S\) is freely suspended from \(O\). In equilibrium the line \(OY\) is inclined at an angle arctan(17) to the horizontal.

1. Find \(r\) in terms of \(h\). [6]

A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \((l + e)\).

1. Find \(e\) in terms of \(l\). [2]

At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt{gl}\).

1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion. [5]
2. Find the amplitude of the simple harmonic motion. [3]
3. Find the time at which the string first goes slack. [4]