3 A small object \(P\) of mass \(m\) is suspended from a fixed point by a light inextensible string of length l. When \(P\) is displaced and released in a certain way, it oscillates in a vertical plane. The time taken for one complete oscillation is called the period and is denoted by \(\tau\). A student is carrying out experiments with \(P\) and suggests the following formula to model the value of \(\tau\).
\(\tau = \mathrm { cg } \mathrm { a } ^ { \mathrm { a } } \mathrm { l } _ { \mathrm { m } } { } ^ { \gamma }\)
in which

• \(g\) is the acceleration due to gravity,
• \(C\) is a dimensionless constant.
  1. Use dimensional analysis to determine the values of the constants \(\alpha , \beta\) and \(\gamma\).
  2. 1. Determine the effect on the period, according to the model, if the length of the string is then multiplied by 4, all other conditions being unchanged.
     2. Determine the effect on the period, according to the model, if instead the mass of the object is multiplied by 4, all other conditions being unchanged.