5 One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity \(\omega\) it can be shown that the kinetic energy \(E\) of the rod is given by \(E = \frac { 1 } { 2 } I \omega ^ { 2 }\), where \(I\) is a quantity called the moment of inertia of the rod.

1. Deduce the dimensions of \(I\).
2. Given that the rod has mass \(m\) and length \(r\), suggest an expression for \(I\), explaining any additional symbols that you use. A student notices that the formula \(E = \frac { 1 } { 2 } I \omega ^ { 2 }\) looks similar to the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\) for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. \(\frac { \mathrm { d } \omega } { \mathrm { d } t }\) ) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be \(F = I \alpha\), where \(F\) is the force applied to the rod and \(\alpha\) is the resulting angular acceleration.
3. Use dimensional analysis to show that the student's suggestion is incorrect.
4. State the dimensions of a quantity \(x\) for which the equation \(F x = I \alpha\) would be dimensionally consistent.
5. Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct.