Dimensional analysis MI context

2 A rod, of length \(x \mathrm {~m}\) and moment of inertia \(I \mathrm {~kg} \mathrm {~m} ^ { 2 }\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through one end. When the rod is hanging at rest, its lower end receives an impulse of magnitude \(J\) Ns, which is just sufficient for the rod to complete full revolutions. It is thought that there is a relationship between \(J , x , I\), the acceleration due to gravity \(g \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and a dimensionless constant \(k\), such that $$J = k x ^ { \alpha } I ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
Find the values of \(\alpha , \beta\) and \(\gamma\) for which this relationship is dimensionally consistent.
[0pt] [6 marks]

2

1. Find the dimensions of energy. The moment of inertia, \(I\), of a rigid body rotating about a fixed axis is measured in \(\mathrm { kg } \mathrm { m } ^ { 2 }\).
2. State the dimensions of \(I\). The kinetic energy, \(E\), of a rigid body rotating about a fixed axis is given by the formula
   \(\mathrm { E } = \frac { 1 } { 2 } \mathrm { I } \omega ^ { 2 }\),
   where \(\omega\) is the angular velocity (angle per unit time) of the rigid body.
3. Show that the formula for \(E\) is dimensionally consistent. When a rigid body is pivoted from one of its end points and allowed to swing freely, it forms a pendulum. The period, \(t\), of the pendulum is the time taken for it to complete one oscillation. A student conjectures the formula
   \(\mathrm { t } = \left. \mathrm { k } ( \mathrm { mg } ) ^ { \alpha } \mathrm { r } ^ { \beta } \right| ^ { \gamma }\),
   where
   • \(k\) is a dimensionless constant,
   • \(m\) is the mass of the rigid body,
   • \(g\) is the acceleration due to gravity,
   • \(r\) is the distance between the pivot point and the rigid body’s centre of mass.
   • Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
   The moment of inertia of a thin uniform rigid rod of mass 1.5 kg and length 0.8 m , rotating about one of its endpoints, is \(0.32 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The student suspends such a rod from one of its endpoints and allows it to swing freely. The student measures the period of this pendulum and finds that it is 1.47 seconds.
4. Using the formula conjectured by the student, determine the value of \(k\).

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.

3 The kinetic energy, \(E\), of a compound pendulum is given by $$E = \frac { 1 } { 2 } I \omega ^ { 2 }$$ where \(\omega\) is the angular speed and \(I\) is a quantity called the moment of inertia.
3

1. Show that for this formula to be dimensionally consistent then \(I\) must have dimensions \(M L ^ { 2 }\), where \(M\) represents mass and \(L\) represents length.
   [0pt] [2 marks]
   3
2. The time, \(T\), taken for one complete swing of a pendulum is thought to depend on its moment of inertia, \(I\), its weight, \(W\), and the distance, \(h\), of the centre of mass of the pendulum from the point of suspension. The formula being proposed is $$T = k I ^ { \alpha } W ^ { \beta } h ^ { \gamma }$$ where \(k\) is a dimensionless constant. Determine the values of \(\alpha , \beta\) and \(\gamma\).