Block on rough horizontal surface – accelerating (finding acceleration or applied force)

1 Fig. 1 shows a block of mass \(M \mathrm {~kg}\) being pushed over level ground by means of a light rod. The force, \(T \mathrm {~N}\), this exerts on the block is along the line of the rod. The ground is rough.
The rod makes an angle \(\alpha\) with the horizontal. \begin{figure}[h] \end{figure}

1. Draw a diagram showing all the forces acting on the block.
2. You are given that \(M = 5 , \alpha = 60 ^ { \circ } , T = 40\) and the acceleration of the block is \(1.5 \mathrm {~ms} ^ { - 2 }\). Find the frictional force.

3 A block \(B\) of mass 0.8 kg is pulled across a horizontal surface by a force of 6 N inclined at an angle of \(60 ^ { \circ }\) to the upward vertical. The coefficient of friction between the block and the surface is 0.2 . Calculate

1. the vertical component of the force exerted on \(B\) by the surface,
2. the acceleration of \(B\). The 6 N force is removed when \(B\) has speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Calculate the time taken for \(B\) to decelerate from a speed of \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to rest.

2. \begin{figure}[h] \end{figure} A particle \(P\) has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude \(F\) newtons, as shown in Figure 1.

1. Find the magnitude of the normal reaction of the plane on \(P\) The particle is accelerating along the plane at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(F\) The coefficient of friction between \(P\) and the plane is \(\mu\)
3. Find the value of \(\mu\), giving your answer to 2 significant figures.

4 Rory pushes a box of mass 2.8 kg across a rough horizontal floor against a resistance of 19 N . Rory applies a constant horizontal force. The box accelerates from rest to \(1.2 \mathrm {~ms} ^ { - 1 }\) as it travels 1.8 m .

1. Calculate the acceleration of the box.
2. Find the magnitude of the force that Rory applies.

16 A block of mass \(m\) kg rests on rough horizontal ground. The coefficient of friction between the block and the ground is \(\mu\). A force of magnitude \(T \mathrm {~N}\) is applied at an angle \(\theta\) radians above the horizontal as shown in the diagram and the block slides without tilting or lifting. \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-10_291_707_388_239}

1. Show that the acceleration of the block is given by \(\frac { T } { m } \cos \theta - \mu g + \frac { T } { m } \mu \sin \theta\). For a fixed value of \(T\), the acceleration of the block depends on the value of \(\theta\). The acceleration has its greatest value when \(\theta = \alpha\).
2. Find an expression for \(\alpha\) in terms of \(\mu\).

3 The diagram shows a rope that is attached to a box of mass 25 kg , which is being pulled along rough horizontal ground. The rope is at an angle of \(30 ^ { \circ }\) to the ground. The tension in the rope is 40 N . The box accelerates at \(0.1 \mathrm {~ms} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_214_729_504_644}

1. Draw a diagram to show all of the forces acting on the box.
2. Show that the magnitude of the friction force acting on the box is 32.1 N , correct to three significant figures.
3. Show that the magnitude of the normal reaction force that the ground exerts on the box is 225 N .
4. Find the coefficient of friction between the box and the ground.
5. State what would happen to the magnitude of the friction force if the angle between the rope and the horizontal were increased. Give a reason for your answer.

2 A block, of mass 4 kg , is made to move in a straight line on a rough horizontal surface by a horizontal force of 50 newtons, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-2_113_1075_913_486} Assume that there is no air resistance acting on the block.

1. Draw a diagram to show all the forces acting on the block.
2. Find the magnitude of the normal reaction force acting on the block.
3. The acceleration of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the friction force acting on the block.
4. Find the coefficient of friction between the block and the surface.
5. Explain how and why your answer to part (d) would change if you assumed that air resistance did act on the block.

8 The diagram shows a block, of mass 20 kg , being pulled along a rough horizontal surface by a rope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-16_323_1194_411_424} The coefficient of friction between the block and the surface is \(\mu\). Model the block as a particle which slides on the surface.

1. If the tension in the rope is 60 newtons, the block moves at a constant speed.
   1. Show that the magnitude of the normal reaction force acting on the block is 166 N .
   2. Find \(\mu\).
2. If the rope remains at the same angle and the block accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the tension in the rope. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-18_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-19_2488_1719_219_150}

2 A wooden block, of mass 4 kg , is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.3 . A horizontal force, of magnitude 30 newtons, acts on the block and causes it to accelerate. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-2_111_771_1146_639}

1. Draw a diagram to show all the forces acting on the block.
2. Calculate the magnitude of the normal reaction force acting on the block.
3. Find the magnitude of the friction force acting on the block.
4. Find the acceleration of the block.

6 A child pulls a sledge, of mass 8 kg , along a rough horizontal surface, using a light rope. The coefficient of friction between the sledge and the surface is 0.3 . The tension in the rope is \(T\) newtons. The rope is kept at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-4_273_775_516_644} Model the sledge as a particle.

1. Draw a diagram to show all the forces acting on the sledge.
2. Find the magnitude of the normal reaction force acting on the sledge, in terms of \(T\).
3. Given that the sledge accelerates at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find \(T\).

7 A block of mass 30 kg is dragged across a rough horizontal surface by a rope that is at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the surface is 0.4 .

1. The tension in the rope is 150 newtons.
   1. Draw a diagram to show the forces acting on the block as it moves.
   2. Show that the magnitude of the normal reaction force on the block is 243 newtons, correct to three significant figures.
   3. Find the magnitude of the friction force acting on the block.
   4. Find the acceleration of the block.
2. When the block is moving, the tension is reduced so that the block moves at a constant speed, with the angle between the rope and the horizontal unchanged. Find the tension in the rope when the block is moving at this constant speed.
3. If the block were made to move at a greater constant speed, again with the angle between the rope and the horizontal unchanged, how would the tension in this case compare to the tension found in part (b)?

6 A floor polisher consists of a heavy metal block with a polishing cloth attached to the underside. A light rod is also attached to the block and is used to push the block across the floor that is to be polished. The block has mass 5 kg . Assume that the floor is horizontal. Model the block as a particle. The coefficient of friction between the cloth and the floor is 0.2 .
A person pushes the rod to exert a force on the block. The force is at an angle of \(60 ^ { \circ }\) to the horizontal and the block accelerates at \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The diagram shows the block and the force exerted by the rod in this situation. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-14_309_205_772_1009} The rod exerts a force of magnitude \(T\) newtons on the block.

1. Find, in terms of \(T\), the magnitude of the normal reaction force acting on the block.
2. \(\quad\) Find \(T\).
   [0pt] [6 marks]

14 A car has an initial velocity of \(1 \mathrm {~ms} ^ { - 1 }\) A particle, \(Q\), moves in a straight line across a rough horizontal surface.
A horizontal driving force of magnitude \(D\) newtons acts on \(Q\) \(Q\) moves with a constant acceleration of \(0.91 \mathrm {~ms} ^ { - 2 }\) \(Q\) has a weight of 0.65 N
The only resistance force acting on \(Q\) is due to friction.
The coefficient of friction between \(Q\) and the surface is 0.4 Find \(D\)

A force of magnitude \(F\) N is applied to a block of mass \(M\) kg which is initially at rest on a horizontal plane. The block starts to move with acceleration 3 ms\(^{-2}\). Modelling the block as a particle, \includegraphics{figure_4}

1. if the plane is smooth, find an expression for \(F\) in terms of \(M\). [2 marks]

If the plane is rough, and the coefficient of friction between the block and the plane is \(\mu\),

1. express \(F\) in terms of \(M\), \(\mu\) and \(g\). [2 marks]
2. Calculate the value of \(\mu\) if \(F = \frac{1}{2}Mg\). [3 marks]