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

3.03m Equilibrium: sum of resolved forces = 03.03n Equilibrium in 2D: particle under forces
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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.
OCR H240/03 2021 November Q10
6 marks Standard +0.3
\includegraphics{figure_10} A block \(D\) of weight 50 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 15 N is applied to \(D\) at an angle \(\theta\) to the horizontal (see diagram).
  1. Complete the diagram in the Printed Answer Booklet showing all the forces acting on \(D\). [1]
It is given that \(D\) remains at rest and the coefficient of friction between \(D\) and the surface is 0.2.
  1. Show that $$15\cos\theta - 3\sin\theta \leqslant 10.$$ [5]