5 A simple pendulum consists of a small sphere of mass \(m\) connected to one end of a light rod of length \(h\). The other end of the rod is freely hinged at a fixed point. When the sphere is pulled a short distance to one side and released from rest the pendulum performs oscillations. The time taken to perform one complete oscillation is called the period and is denoted by \(P\).

1. Assuming that \(P = k m ^ { \alpha } h ^ { \beta } g ^ { \gamma }\), where \(g\) is the acceleration due to gravity and \(k\) is a dimensionless constant, find the values of \(\alpha , \beta\) and \(\gamma\). A student conducts an experiment to investigate how \(P\) varies as \(h\) varies. She measures the value of \(P\) for various values of \(h\), ensuring that all other conditions remain constant. Her results are summarised in the table below.

   \(h ( \mathrm {~m} )\)	0.40	2.50	3.60
   \(P ( \mathrm {~s} )\)	1.27	2.17	3.81

2. Show that these results are not consistent with the answers to part (i).
3. The student later realises that she has recorded one of her values of \(P\) incorrectly.
   • Identify the incorrect value.
   • Estimate the correct value that she should have recorded.