Minimum/maximum force for equilibrium

A question is this type if and only if it requires finding the minimum or maximum value of an applied force (or tension) to maintain equilibrium or cause motion, given friction constraints.

2 questions · Standard +0.6

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Edexcel M1 2018 June Q2
10 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-04_333_976_287_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of mass 2 kg lies on a rough plane. The plane is inclined to the horizontal at \(30 ^ { \circ }\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\). The particle is held in equilibrium by a force of magnitude \(P\) newtons. The force makes an angle of \(20 ^ { \circ }\) with the horizontal and acts in a vertical plane containing a line of greatest slope of the plane, as shown in Figure 1. Find the least possible value of \(P\).
Edexcel M1 2022 October Q3
11 marks Standard +0.3
\includegraphics{figure_2} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) A particle \(P\) of mass 2 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.
  1. Show that when \(X = 14.7\) there is no frictional force acting on \(P\) [3] The coefficient of friction between \(P\) and the plane is 0.5
  2. Find the smallest possible value of \(X\). [8]