IP2 Class Test (May 2025) Q6

In a physics experiment, a formula is given by \(F = mg \sin(\theta) - \mu mg \cos(\theta)\), where \(\mu\), \(g\) are constants and \(m\), \(\theta\) may vary. It is given that \(g = 9.8\), \(\mu = 0.5\), \(m > 0\) and \(0 < \theta < 90{}^\circ\).

1. Evaluate \(F\) given that \(m = 15\) and \(\theta = 30{}^\circ\).[1]
2. Find the value of \(\theta\) such that \(F = 0\). Note that \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).[2]
3. Another variable, \(a\), is defined by \(a = \frac{F}{m}\). Express \(a\) in terms of \(\mu\), \(g\) and \(\theta\).
   Hence find the range of values of \(\theta\) such that \(a > 0\).
   You may assume the fact that \(\tan(\theta) > p \iff \theta > \tan^{-1}(p)\) for \(0 < \theta < 90{}^\circ\).[2]