Practical friction scenarios

7 Clive and Ken are trying to move a box of mass 50 kg on a rough, horizontal floor. As shown in Fig. 7, Clive always pushes horizontally and Ken always pulls at an angle of \(30 ^ { \circ }\) to the horizontal. Each of them applies forces to the box in the same vertical plane as described below. \begin{figure}[h] \end{figure} Initially, the box is in equilibrium with Clive pushing with a force of 60 N and Ken not pulling at all.

1. What is the resistance to motion of the box? Ken now adds a pull of 70 N to Clive's push of 60 N . The box remains in equilibrium.
2. What now is the resistance to motion of the box?
3. Calculate the normal reaction of the floor on the box. The frictional resistance to sliding of the box is 125 N .
   Clive now pushes with a force of 160 N but Ken does not pull at all.
4. Calculate the acceleration of the box. Clive stops pushing when the box has a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. How far does the box then slide before coming to rest? Ken and Clive now try again. Ken pulls with a force of \(Q \mathrm {~N}\) and Clive pushes with a force of 160 N . The frictional resistance to sliding of the box is now 115 N and the acceleration of the box is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
6. Calculate the value of \(Q\).

2 In the sport of curling, a heavy stone is projected across a horizontal ice surface. One player projects a stone of weight 180 N , which moves 36 m in a straight line and comes to rest 24 s after the instant of projection. The only horizontal force acting on the stone after its projection is a constant frictional force between the stone and the ice.

1. Calculate the deceleration of the stone.
2. Find the magnitude of the frictional force acting on the stone, and calculate the coefficient of friction between the stone and the ice.

7 An explorer is trying to pull a loaded sledge of total mass 100 kg along horizontal ground using a light rope. The only resistance to motion of the sledge is from friction between it and the ground. \begin{figure}[h] \end{figure} Initially she pulls with a force of 121 N on the rope inclined at \(34 ^ { \circ }\) to the horizontal, as shown in Fig. 7, but the sledge does not move.

1. Draw a diagram showing all the forces acting on the sledge. Show that the frictional force between the ground and the sledge is 100 N , correct to 3 significant figures. Calculate the normal reaction of the ground on the sledge. The sledge is given a small push to set it moving at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The explorer continues to pull on the rope with the same force and the same angle as before. The frictional force is also unchanged.
2. Describe the subsequent motion of the sledge. The explorer now pulls the rope, still at an angle of \(34 ^ { \circ }\) to the horizontal, so that the tension in it is 155 N . The frictional force is now 95 N .
3. Calculate the acceleration of the sledge. In a new situation, there is no rope and the sledge slides down a uniformly rough slope inclined at \(26 ^ { \circ }\) to the horizontal. The sledge starts from rest and reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 2 seconds.
4. Calculate the frictional force between the slope and the sledge.

8 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-5_502_935_1411_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
5. Find the velocity of the tub when it has moved a further 1.65 m .

   4	□ box P □
   \multirow[t]{10}{*}{4 }
   4

5 A woman is pulling a loaded sledge along horizontal ground. The only resistance to motion of the sledge is due to friction between it and the ground. \begin{figure}[h] \end{figure} At first, she pulls with a force of 100 N inclined at \(32 ^ { \circ }\) to the horizontal, as shown in Fig.5, but the sledge does not move.

1. Determine the frictional force between the ground and the sledge. Give your answer correct to 3 significant figures.
2. Next she pulls with a force of 100 N inclined at a smaller angle to the horizontal. The sledge still does not move. Compare the frictional force in this new situation with that in part (a), justifying your answer.

\includegraphics{figure_1} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at 30° to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is \(\frac{1}{5}\). By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope. [8]

A lifeboat slides down a straight ramp inclined at an angle of \(15°\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from rest at the top of the ramp and is moving with a speed of 12.6 m s\(^{-1}\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp. [9]

A sledge of mass 4 kg rests in limiting equilibrium on a rough slope inclined at an angle 10° to the horizontal. By modelling the sledge as a particle,

1. show that the coefficient of friction, \(\mu\), between the sledge and the ground is 0.176 correct to 3 significant figures. [6 marks]

The sledge is placed on a steeper part of the slope which is inclined at an angle 30° to the horizontal. The value of \(\mu\) remains unchanged.

1. Find the minimum extra force required along the line of greatest slope to prevent the sledge from slipping down the hill. [5 marks]