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 \(B\) 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
the work done by \(\mathbf { F }\) as the bead moves from \(A\) to \(B\),
the speed of the bead at \(B\).
(a) Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 a\), about an axis perpendicular to the rod through its centre is \(\frac { 1 } { 3 } m a ^ { 2 }\).
(3)
A uniform wire of mass \(4 m\) and length \(8 a\) is bent into the shape of a square.
Find the moment of inertia of the square about the axis through the centre of the square perpendicular to its plane.
(4)