Limiting equilibrium with variable angle

A question is this type if and only if it involves finding a critical angle (of applied force or incline) at which limiting equilibrium occurs or motion begins.

1 questions · Standard +0.3

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Pre-U Pre-U 9794/3 2013 June Q9
9 marks Standard +0.3
9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}
  1. Copy and complete the diagram to show all the forces acting on the particle.
  2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
  3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.