3.03u Static equilibrium: on rough surfaces

1 \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-2_200_537_269_804} A box of weight 100 N rests in equilibrium on a plane inclined at an angle \(\alpha\) to the horizontal. It is given that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). A force of magnitude \(P \mathrm {~N}\) acts on the box parallel to the plane in the upwards direction (see diagram). The coefficient of friction between the box and the plane is 0.25 .

1. Find the magnitude of the normal force acting on the box.
2. Given that the equilibrium is limiting, show that the magnitude of the frictional force acting on the box is 24 N .
3. Find the value of \(P\) for which the box is on the point of slipping
   1. down the plane,
   2. up the plane.

6 A block of weight 14.7 N is at rest on a horizontal floor. A force of magnitude 4.9 N is applied to the block.

1. The block is in limiting equilibrium when the 4.9 N force is applied horizontally. Show that the coefficient of friction is \(\frac { 1 } { 3 }\).
2. 
   
   [diagram]
   When the force of 4.9 N is applied at an angle of \(30 ^ { \circ }\) above the horizontal, as shown in the diagram, the block moves across the floor. Calculate
   1. the vertical component of the contact force between the floor and the block, and the magnitude of the frictional force,
   2. the acceleration of the block.
   3. Calculate the magnitude of the frictional force acting on the block when the 4.9 N force acts at an angle of \(30 ^ { \circ }\) to the upward vertical, justifying your answer fully.

4 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_200_897_269_625} A block of mass 3 kg is placed on a horizontal surface. A force of magnitude 20 N acts downwards on the block at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).

1. Given that the surface is smooth, calculate the acceleration of the block.
2. Given instead that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the surface.

5 \includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-3_697_579_1238_781} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(A P\) and \(B P\) of the string are taut. The system is in equilibrium with angle \(B A P = \alpha\) and angle \(A B P = \beta\) (see diagram). The weight of \(A\) is 2 N and the tensions in the parts \(A P\) and \(B P\) of the string are 7 N and \(T \mathrm {~N}\) respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.

1. Find the coefficient of friction between the wire and the ring \(A\).
2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\).
3. Given that there is no frictional force acting on \(B\), find the mass of \(B\).

5 A block of mass \(m \mathrm {~kg}\) is at rest on a horizontal plane. The coefficient of friction between the block and the plane is 0.2 .

1. When a horizontal force of magnitude 5 N acts on the block, the block is on the point of slipping. Find the value of \(m\).
2. \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-3_312_711_1244_758} When a force of magnitude \(P \mathrm {~N}\) acts downwards on the block at an angle \(\alpha\) to the horizontal, as shown in the diagram, the frictional force on the block has magnitude 6 N and the block is again on the point of slipping. Find
   1. the value of \(\alpha\) in degrees,
   2. the value of \(P\).

7 A particle of mass 0.1 kg is at rest at a point \(A\) on a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The particle is given an initial velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .

1. Find the coefficient of friction between the particle and the plane.
2. Show that, after coming to instantaneous rest, the particle moves down the plane.
3. Find the speed with which the particle passes through \(A\) during its downward motion.

7 \begin{figure}[h] \end{figure} A rectangular block \(B\) of weight 12 N lies in limiting equilibrium on a horizontal surface. A horizontal force of 4 N and a coplanar force of 5 N inclined at \(60 ^ { \circ }\) to the vertical act on \(B\) (see Fig. 1).

1. Find the coefficient of friction between \(B\) and the surface. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(B\) is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is \(\mu\).
2. Given that the upper block is in limiting equilibrium, find \(\mu\).
3. Given instead that \(\mu = 0.1\), find the accelerations of the two blocks.

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 \(Q\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 4 m and weight 50 N .
The rod has its end \(A\) on rough horizontal ground. The rod is held in equilibrium at an angle \(\alpha\) to the ground by a light inextensible cable attached to the rod at \(B\), as shown in Figure 2. The cable and the rod lie in the same vertical plane and the cable is perpendicular to the rod. The tension in the cable is \(T\) newtons. Given that \(\sin \alpha = \frac { 3 } { 5 }\)

1. show that \(T = 20\) Given also that the rod is in limiting equilibrium,
2. find the value of the coefficient of friction between the rod and the ground.

5. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of mass 15 kg and length 6 m , rests with end \(A\) on rough horizontal ground. The end \(B\) of the beam rests against a rough vertical wall. The beam is inclined at \(75 ^ { \circ }\) to the ground, as shown in Figure 2.
The coefficient of friction between the beam and the wall is 0.2
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall. The beam rests in limiting equilibrium.

1. Find the magnitude of the normal reaction between the beam and the wall at \(B\).
2. Find the value of \(\mu\)

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of mass \(m\) and length \(2 a\), rests in limiting equilibrium with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall.
The vertical plane containing the rod is at right angles to the wall.
The rod is inclined to the wall at an angle \(\alpha\), as shown in Figure 2.
The coefficient of friction between the rod and the ground is \(\frac { 1 } { 3 }\)

1. Show that \(\tan \alpha = \frac { 2 } { 3 }\) With the rod in the same position, a horizontal force of magnitude \(k m g\) is applied to the \(\operatorname { rod }\) at \(A\), towards the wall. The line of action of this force is at right angles to the wall. The rod remains in equilibrium.
2. Find the largest possible value of \(k\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of length 8 m and weight \(W\) newtons rests in equilibrium against a rough horizontal peg \(P\). The end \(A\) is on rough horizontal ground. The friction is limiting at both \(A\) and \(P\). The distance \(A P\) is 5 m , as shown in Figure 1. The rod rests at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The rod is in a vertical plane which is perpendicular to \(P\). The coefficient of friction between the rod and \(P\) is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the ground is \(\mu\).

1. Show that the magnitude of the normal reaction between the rod and \(P\) is \(0.48 W\) newtons.
2. Find the value of \(\mu\).

4. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The end \(A\) rests against a rough vertical wall. One end of a light inextensible string is attached to the rod at the point \(C\), where \(A C = 2 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is vertically above \(A\), with the string perpendicular to the rod. A particle of mass \(m\) is attached to the rod at the end \(B\). The rod is in equilibrium in a vertical plane which is perpendicular to the wall. The rod is inclined at \(60 ^ { \circ }\) to the wall, as shown in Figure 1. Find, in terms of \(m\) and \(g\),

1. the tension in the string,
2. the magnitude of the horizontal component of the force exerted by the wall on the rod. The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
3. find the value of \(\mu\).

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of mass \(8 m\) and length \(2 a\), has its end \(A\) resting against a rough vertical wall. One end of a light inextensible string is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\), which is vertically above \(A\). The angle between the rod and the string is \(30 ^ { \circ }\). A particle of mass \(k m\) is fixed to the rod at \(C\), where \(A C = 0.5 a\). The rod is in equilibrium in a vertical plane perpendicular to the wall, and is at an angle of \(60 ^ { \circ }\) to the wall, as shown in Figure 5. The tension in the string is \(T\).

1. Show that \(T = \frac { \sqrt { 3 } } { 4 } ( 16 + k ) m g\) The coefficient of friction between the wall and the rod is \(\frac { 2 } { 3 } \sqrt { 3 }\).
   Given that the rod is in limiting equilibrium,
2. find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-23_67_65_2656_1886}

5. \begin{figure}[h] \end{figure} A pole \(A B\) has length 2.5 m and weight 70 N .
The pole rests with end \(B\) against a rough vertical wall. One end of a cable of length 4 m is attached to the pole at \(A\). The other end of the cable is attached to the wall at the point \(C\). The point \(C\) is vertically above \(B\) and \(B C = 2.5 \mathrm {~m}\).
The angle between the cable and the wall is \(\alpha\), as shown in Figure 2.
The pole is in a vertical plane perpendicular to the wall.
The cable is modelled as a light inextensible string and the pole is modelled as a uniform rod. Given that \(\tan \alpha = \frac { 3 } { 4 }\)

1. show that the tension in the cable is 56 N . Given also that the pole is in limiting equilibrium,
2. find the coefficient of friction between the pole and the wall. \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-15_90_61_2613_1886}

2. \begin{figure}[h] \end{figure} Figure 1 shows a ladder \(A B\), of mass 25 kg and length 4 m , resting in equilibrium with one end \(A\) on rough horizontal ground and the other end \(B\) against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is \(\frac { 11 } { 25 }\). The ladder makes an angle \(\beta\) with the ground. When Reece, who has mass 75 kg , stands at the point \(C\) on the ladder, where \(A C = 2.8 \mathrm {~m}\), the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle.

1. Find the magnitude of the frictional force of the ground on the ladder.
2. Find, to the nearest degree, the value of \(\beta\).
3. State how you have used the modelling assumption that Reece is a particle.

4. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 5 } { 8 }\). When each particle has moved a distance \(h , B\) has not reached the ground and \(A\) has not reached \(P\).

1. Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the workenergy principle,
2. find an expression for \(v ^ { 2 }\), giving your answer in the form \(k g h\), where \(k\) is a number.

5. \begin{figure}[h] \end{figure} A plank rests in equilibrium against a fixed horizontal pole. The plank is modelled as a uniform rod \(A B\) and the pole as a smooth horizontal peg perpendicular to the vertical plane containing \(A B\). The rod has length \(3 a\) and weight \(W\) and rests on the peg at \(C\), where \(A C = 2 a\). The end \(A\) of the rod rests on rough horizontal ground and \(A B\) makes an angle \(\alpha\) with the ground, as shown in Figure 2.

1. Show that the normal reaction on the rod at \(A\) is \(\frac { 1 } { 4 } \left( 4 - 3 \cos ^ { 2 } \alpha \right) W\). Given that the rod is in limiting equilibrium and that \(\cos \alpha = \frac { 2 } { 3 }\),
2. find the coefficient of friction between the rod and the ground.

3. \begin{figure}[h] \end{figure} A non-uniform rod, \(A B\), of mass \(m\) and length 2l, rests in equilibrium with one end \(A\) on a rough horizontal floor and the other end \(B\) against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of \(60 ^ { \circ }\) with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the wall is \(\frac { 2 } { 3 }\). The rod is on the point of slipping at both ends.

1. Find the magnitude of the vertical component of the force exerted on the rod by the floor. The centre of mass of the rod is at \(G\).
2. Find the distance \(A G\).

1. A ladder \(A B\), of weight \(W\) and length \(2 l\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a rough vertical wall. The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\). The coefficient of friction between the ladder and the ground is \(\mu\). Friction is limiting at both \(A\) and \(B\). The ladder is at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 5 } { 3 }\). The ladder is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall.

Find the value of \(\mu\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 5 kg and length 8 m , has its end \(B\) resting on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), by a rope attached to the rod at \(C\). The distance \(A C = 1 \mathrm {~m}\). The rope is in the same vertical plane as the rod. The angle between the rope and the rod is \(\beta\) and the tension in the rope is \(T\) newtons, as shown in Figure 3. The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\). The vertical component of the force exerted on the rod at \(B\) by the ground is \(R\) newtons.

1. Find the value of \(R\).
2. Find the size of angle \(\beta\).

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), rests with its end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a light string attached to the rod at \(B\), as shown in Figure 3. The string is perpendicular to the rod and lies in the same vertical plane as the rod. The coefficient of friction between the ground and the rod is \(\mu\).
Show that \(\mu = \frac { \cos \theta \sin \theta } { 2 - \cos ^ { 2 } \theta }\)

4. \section*{Figure 1} A uniform ladder, of mass \(m\) and length \(2 a\), has one end on rough horizontal ground. The other end rests against a smooth vertical wall. A man of mass \(3 m\) stands at the top of the ladder and the ladder is in equilibrium. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\), and the ladder makes an angle \(\alpha\) with the vertical, as shown in Fig. 1. The ladder is in a vertical plane perpendicular to the wall. Show that \(\tan \alpha \leq \frac { 2 } { 7 }\).

1. A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string, of natural length 0.8 m and modulus of elasticity 0.6 N . The other end of the string is fixed to a point \(A\) on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 7 }\)

The particle \(P\) is projected from \(A\), with speed \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface of the table.
After travelling 0.8 m from \(A\), the particle passes through the point \(B\) on the table.

1. Find the speed of \(P\) at the instant it passes through \(B\). The particle \(P\) comes to rest at the point \(C\) on the table, where \(A B C\) is a straight line.
2. Find the total distance travelled by \(P\) as it moves directly from \(A\) to \(C\).
3. Show that \(P\) remains at rest at \(C\).