Maximum/minimum force for equilibrium

A question is this type if and only if it asks for the greatest, least, or limiting value of a force or angle for which a particle remains in equilibrium, often involving limiting friction or geometric constraints.

3 questions · Standard +0.5

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CAIE M1 2021 November Q6
8 marks Standard +0.3
6
\includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-08_412_588_260_776} A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal force of magnitude \(X \mathrm {~N}\), as shown in the diagram. The two strings are both inclined at \(60 ^ { \circ }\) to the vertical.
  1. Given that \(X = 100\), find the tension in the lower string.
  2. Find the least value of \(X\) for which the block remains in equilibrium in the position shown. [4]
Edexcel M1 2021 January Q3
9 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ca445c1e-078c-4a57-94df-de90f30f8efd-06_156_1009_255_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A parcel of mass 20 kg is at rest on a rough horizontal floor. The coefficient of friction between the parcel and the floor is 0.3 Two forces, both acting in the same vertical plane, of magnitudes 200 N and \(T \mathrm {~N}\) are applied to the parcel. The line of action of the 200 N force makes an angle of \(15 ^ { \circ }\) with the horizontal and the line of action of the \(T \mathrm {~N}\) force makes an angle of \(25 ^ { \circ }\) with the horizontal, as shown in Figure 1. The parcel is modelled as a particle \(P\). Find the smallest value of \(T\) for which \(P\) remains in equilibrium.
CAIE M1 2022 June Q4
8 marks Standard +0.3
  1. In the case where \(F = 20\), find the tensions in each of the strings.
  2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.