Edexcel M5 (Mechanics 5) Specimen

A bead of mass 0.125 kg is threaded on a smooth straight horizontal wire. The bead moves from rest at the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j} - \mathbf{k})\) m relative to a fixed origin \(O\) to a point with position vector \((3\mathbf{i} - 4\mathbf{j} - \mathbf{k})\) m relative to \(O\) under the action of a force \(\mathbf{F} = (14\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) N. Find

1. the work done by \(\mathbf{F}\) as the bead moves from \(A\) to \(B\), [3]
2. the speed of the bead at \(B\). [2]

1. Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2a\), about an axis perpendicular to the rod through its centre is \(\frac{1}{3}ma^2\). [3]

A uniform wire of mass \(4m\) and length \(8a\) is bent into the shape of a square.

1. Find the moment of inertia of the square about the axis through the centre of the square perpendicular to its plane. [4]

Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) and a couple \(\mathbf{G}\) act on a rigid body. The force \(\mathbf{F}_1 = (3\mathbf{i} + 4\mathbf{j})\) N acts through the point with position vector \(2\mathbf{i}\) m relative to a fixed origin \(O\). The force \(\mathbf{F}_2 = (2\mathbf{i} - \mathbf{j} + \mathbf{k})\) N acts through the point with position vector \((\mathbf{i} + \mathbf{j})\) m relative to \(O\). The forces and couple are equivalent to a single force \(\mathbf{F}\) acting through \(O\).

1. Find \(\mathbf{F}\). [2]
2. Find \(\mathbf{G}\). [5]

A uniform circular disc, of mass \(2m\) and radius \(a\), is free to rotate in a vertical plane about a fixed, smooth horizontal axis through a point of its circumference. The axis is perpendicular to the plane of the disc. The disc hangs in equilibrium. A particle \(P\) of mass \(m\) is moving horizontally in the same plane as the disc with speed \(\sqrt{20ag}\). The particle strikes, and adheres to, the disc at one end of its horizontal diameter.

1. Find the angular speed of the disc immediately after \(P\) strikes it. [7]
2. Verify that the disc will turn through an angle of \(90°\) before first coming to instantaneous rest. [3]

A uniform square lamina \(ABCD\) of side \(a\) and mass \(m\) is free to rotate in vertical plane about a horizontal axis through \(A\). The axis is perpendicular to the plane of the lamina. The lamina is released from rest when \(t = 0\) and \(AC\) makes a small angle with the downward vertical through \(A\).

1. Show that the moment of inertia of the lamina about the axis is \(\frac{2}{3}ma^2\). [3]
2. Show that the motion of the lamina is approximately simple harmonic. [5]
3. Find the time \(t\) when \(AC\) is first vertical. [2]

A uniform rod \(AB\) of mass \(m\) and length \(4a\) is free to rotate in a vertical plane about a horizontal axis through the point \(O\) of the rod, where \(OA = a\). The rod is slightly disturbed from rest when \(B\) is vertically above \(A\).

1. Find the magnitude of the angular acceleration of the rod when it is horizontal. [4]
2. Find the angular speed of the rod when it is horizontal. [2]
3. Calculate the magnitude of the force acting on the rod at \(O\) when the rod is horizontal. [5]

As a hailstone falls under gravity in still air, its mass increases. At time \(t\) the mass of the hailstone is \(m\). The hailstone is modelled as a uniform sphere of radius \(r\) such that $$\frac{dr}{dt} = kr,$$ where \(k\) is a positive constant.

1. Show that \(\frac{dm}{dt} = 3km\). [2]

Assuming that there is no air resistance,

1. show that the speed \(v\) of the hailstone at time \(t\) satisfies $$\frac{dv}{dt} = g - 3kv.$$ [4]

Given that the speed of the hailstone at time \(t = 0\) is \(u\),

1. find an expression for \(v\) in terms of \(t\). [5]
2. Hence show that the speed of the hailstone approaches the limiting value \(\frac{g}{3k}\). [1]

A particle \(P\) moves in the \(x\)-\(y\) plane and has position vector \(\mathbf{r}\) metres relative to a fixed origin \(O\) at time \(t\) s. Given that \(\mathbf{r}\) satisfies the vector differential equation $$\frac{d^2\mathbf{r}}{dt^2} + 9\mathbf{r} = 8\sin t \mathbf{i}$$ and that when \(t = 0\) s, \(P\) is at \(O\) and moving with velocity \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\),

1. find \(\mathbf{r}\) at time \(t\). [11]
2. Hence find when \(P\) next returns to \(O\). [2]