Block on horizontal plane motion

5 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-08_286_661_260_742} A block of mass 5 kg is being pulled along a rough horizontal floor by a force of magnitude \(X \mathrm {~N}\) acting at \(30 ^ { \circ }\) above the horizontal (see diagram). The block starts from rest and travels 2 m in the first 5 s of its motion.

1. Find the acceleration of the block.
2. Given that the coefficient of friction between the block and the floor is 0.4 , find \(X\).
   The block is now placed on a part of the floor where the coefficient of friction between the block and the floor has a different value. The value of \(X\) is changed to 25, and the block is now in limiting equilibrium.
3. Find the value of the coefficient of friction between the block and this part of the floor.

1 A block is at rest on a rough horizontal plane. The coefficient of friction between the block and the plane is 1.25 .

1. State, giving a reason for your answer, whether the minimum vertical force required to move the block is greater or less than the minimum horizontal force required to move the block. A horizontal force of continuously increasing magnitude \(P \mathrm {~N}\) and fixed direction is applied to the block.
2. Given that the weight of the block is 60 N , find the value of \(P\) when the acceleration of the block is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

4 A particle \(P\) of mass 0.8 kg is placed on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\mu\). A force of magnitude 5 N , acting upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), is applied to \(P\). The particle is on the point of sliding on the table.

1. Find the value of \(\mu\).
2. The magnitude of the force acting on \(P\) is increased to 10 N , with the direction of the force remaining the same. Find the acceleration of \(P\).

1 A particle of mass 2 kg is initially at rest on a rough horizontal plane. A force of magnitude 10 N is applied to the particle at \(15 ^ { \circ }\) above the horizontal. It is given that 10 s after the force is applied, the particle has a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the magnitude of the frictional force is 8.96 N , correct to 3 significant figures.
2. Find the coefficient of friction between the particle and the plane.

5 A particle \(P\) of mass 0.4 kg is at rest on a horizontal surface. The coefficient of friction between \(P\) and the surface is 0.2 . A force of magnitude 1.2 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal is then applied to \(P\). Find the acceleration of \(P\) in each of the following cases:

1. \(\theta = 0\);
2. \(\theta = 20\);
3. \(\theta = 70\);
4. \(\theta = 90\).

2 A block, of mass 10 kg , is at rest on a rough horizontal surface, when a horizontal force, of magnitude \(P\) newtons, is applied to the block, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-04_108_962_461_539} The coefficient of friction between the block and the surface is 0.5 .

1. Draw and label a diagram to show all the forces acting on the block.
2. 1. Calculate the magnitude of the normal reaction force acting on the block.
   2. Find the maximum possible magnitude of the friction force between the block and the surface.
   3. Given that \(P = 30\), state the magnitude of the friction force acting on the block.
3. Given that \(P = 80\), find the acceleration of the block.
   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-05_2484_1709_223_153}

A block of mass \(2\) kg is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is \(0.3\).

1. Calculate the maximum frictional force that can act on the block. [2]
2. A horizontal force of \(5\) N is applied to the block. Calculate the acceleration of the block. [3]

A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac{24}{25}\). The force is applied for a period of 5 s, during which time the block moves a distance of 4.5 m.

1. Find the magnitude of the frictional force on the block. [4]
2. Show that the coefficient of friction between the block and the plane is 0.165, correct to 3 significant figures. [3]
3. When the block has moved a distance of 4.5 m, the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion. [4]