Single angled force - find limiting friction or coefficient

3 A string is attached to a block of mass 4 kg which rests in limiting equilibrium on a rough horizontal table. The string makes an angle of \(24 ^ { \circ }\) above the horizontal and the tension in the string is 30 N .

1. Draw a diagram showing all the forces acting on the block.
2. Find the coefficient of friction between the block and the table.

1 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_426_424_258_863} A block \(B\) of mass 7 kg is at rest on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts on \(B\) at an angle of \(15 ^ { \circ }\) to the upward vertical (see diagram).

1. Given that \(B\) is in equilibrium find, in terms of \(X\), the normal component of the force exerted on \(B\) by the ground.
2. The coefficient of friction between \(B\) and the ground is 0.4 . Find the value of \(X\) for which \(B\) is in limiting equilibrium.

1 A block of mass 400 kg rests in limiting equilibrium on horizontal ground. A force of magnitude 2000 N acts on the block at an angle of \(15 ^ { \circ }\) to the upwards vertical. Find the coefficient of friction between the block and the ground, correct to 2 significant figures.

4. \begin{figure}[h] \end{figure} A small block of mass 5 kg lies at rest on a rough horizontal plane.
The coefficient of friction between the block and the plane is \(\frac { 3 } { 7 }\) A force of magnitude \(P\) newtons is applied to the block in a direction which makes an angle of \(30 ^ { \circ }\) with the plane, as shown in Figure 1. The block is modelled as a particle.
Given that \(P = 14\)

1. find the magnitude of the frictional force exerted on the block by the plane and describe what happens to the block, justifying your answer.
   (6) The value of \(P\) is now changed so that the block is on the point of slipping along the plane.
2. Find the value of \(P\)

A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]

\includegraphics{figure_5} A particle of mass 12 kg is going to be pulled across a rough horizontal plane by a light inextensible string. The string is at an angle of 30° above the plane and has tension \(T\) N (see diagram). The coefficient of friction between the particle and the plane is 0.5.

1. Given that the particle is on the point of moving, find the value of \(T\). [5]
2. Given instead that the particle is accelerating at 0.2 ms\(^{-2}\), find the value of \(T\). [3]

\includegraphics{figure_1} A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. A force of magnitude \(P\) newtons is applied to the box at 50° to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of \(P\), giving your answer to 2 significant figures. [9]

A string is attached to a packing case of mass 12 kg, which is at rest on a rough horizontal plane. When a force of magnitude 50 N is applied at the other end of the string, and the string makes an angle of 35° with the vertical as shown, the case is on the point of moving. \includegraphics{figure_3}

1. Find the coefficient of friction between the case and the plane. [5 marks]

The force is now increased, with the string at the same angle, and the case starts to move along the plane with constant acceleration, reaching a speed of 2 ms\(^{-1}\) after 4 seconds.

1. Find the magnitude of the new force. [5 marks]
2. State any modelling assumptions you have made about the case and the string. [2 marks]

\includegraphics{figure_2} Figure 2 shows a large block of mass 50 kg being pulled on rough horizontal ground by means of a rope attached to the block. The tension in the rope is 200 N and it makes an angle of 40° with the horizontal. Under these conditions, the block is on the point of moving. Modelling the block as a particle,

1. show that the coefficient of friction between the block and the ground is 0.424 correct to 3 significant figures. [6 marks]

The angle with the horizontal at which the rope is being pulled is reduced to 30°. Ignoring air resistance and assuming that the tension in the rope and the coefficient of friction remain unchanged,

1. find the acceleration of the block. [6 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85

1. The boy applies a horizontal force of 150 N. Show that the crate remains stationary. [3 marks]
2. Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N, at an angle of 15° above the horizontal, as shown in the diagram. \includegraphics{figure_5} Determine whether the crate remains stationary. Fully justify your answer. [5 marks]