- (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 one end is \(\frac { 4 } { 3 } m a ^ { 2 }\).
(b) Hence, or otherwise, find the moment of inertia of a uniform square lamina, of mass \(M\) and side \(2 a\), about an axis through one corner and perpendicular to the plane of the lamina. - A particle of mass 0.5 kg is at rest at the point with position vector ( \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) ) m. The particle is then acted upon by two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \(( 4 \mathbf { i } + 5 \mathbf { j } - 5 \mathbf { k } ) \mathrm { m }\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\), find \(\mathbf { F } _ { 2 }\).
- A particle \(P\) moves in the \(x - y\) plane and has position vector \(\mathbf { r }\) metres at time \(t\) seconds. It is given that \(\mathbf { r }\) satisfies the differential equation
$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t }$$
When \(t = 0 , P\) is at the point with position vector \(3 \mathbf { i }\) metres and is moving with velocity \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).
(a) Find \(\mathbf { r }\) in terms of \(t\).
(b) Describe the path of \(P\), giving its cartesian equation.