3.03i Normal reaction force

1 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

6 An empty open box of mass 4 kg is on a plane that is inclined at \(25 ^ { \circ }\) to the horizontal.
In one model the plane is taken to be smooth. The box is held in equilibrium by a string with tension \(T \mathrm {~N}\) parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h] \end{figure}

1. Calculate \(T\). A rock of mass \(m \mathrm {~kg}\) is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
2. Find \(m\). In a refined model the plane is rough. The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of \(P \mathrm {~N}\) inclined at \(15 ^ { \circ }\) to the plane, as shown in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_359_559_1599_830} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Draw a diagram showing all the forces acting on the box.
4. Calculate \(P\).
5. Calculate the normal reaction of the plane on the box.

7 A block of mass 4 kg is in equilibrium on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4. A frictional force of 10 N acts up the plane and a vertical string AB attached to the block is in tension. \begin{figure}[h] \end{figure}

1. Draw a diagram showing the four forces acting on the block.
2. By considering the components of the forces parallel to the slope, calculate the tension in the string.
3. Calculate the normal reaction of the plane on the block.

5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm . The other end of the string is attached to a fixed point \(A\) that is vertically above the point \(O\) on a smooth horizontal table, such that \(O A = 40 \mathrm {~cm}\). The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre \(O\), as shown in Figure 4. The particle is moving with a constant angular speed of 3 radians per second.

1. Find (i) the tension in the string,
   (ii) the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
2. Find the angular speed of \(P\) at the instant \(P\) loses contact with the table.

6 A box, of mass 3 kg , is placed on a slope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.

1. A simple model assumes that the slope is smooth.
   1. Draw a diagram to show the forces acting on the box.
   2. Show that the acceleration of the box is \(4.9 \mathrm {~ms} ^ { - 2 }\).
2. A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
   1. Show that the acceleration of the box is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the magnitude of the friction force acting on the box.
   3. Find the coefficient of friction between the box and the slope.
   4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.

9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}

1. Copy and complete the diagram to show all the forces acting on the particle.
2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.

\includegraphics{figure_2} A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20°\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find

1. the normal reaction of the plane on the box, [3]
2. the acceleration of the box. [5]

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]

\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]

In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.

1. Show that \(\mu = 0.83\) [4 marks]
2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
   1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
   2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]

A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\). For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4. \includegraphics{figure_4}

1. By considering the energy of P, show that \(v^2 = u^2 + 2ga(1 - \cos\theta)\). [2]
2. Show that the magnitude of the normal contact force between the sphere and particle P is $$mg(3\cos\theta - 2) - \frac{mv^2}{a}.$$ [2]

The particle loses contact with the sphere when \(\cos\theta = \frac{3}{4}\).

1. Find an expression for \(u\) in terms of \(a\) and \(g\). [2]