3.03m Equilibrium: sum of resolved forces = 0

2. \begin{figure}[h] \end{figure} A particle \(P\) of weight 40 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 20 N is applied to \(P\). The force acts at angle \(\theta\) to the horizontal, as shown in Figure 2. The coefficient of friction between \(P\) and the surface is \(\mu\). Given that the particle remains at rest, show that $$\mu \geqslant \frac { \cos \theta } { 2 + \sin \theta }$$ \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-07_119_167_2615_1777}

5. \begin{figure}[h] \end{figure} A small metal box of mass 6 kg is attached at \(B\) to two ropes \(B P\) and \(B Q\). The fixed points \(P\) and \(Q\) are on a horizontal ceiling and \(P Q = 3.5 \mathrm {~m}\). The box hangs in equilibrium at a vertical distance of 2 m below the line \(P Q\), with the ropes in a vertical plane and with angle \(B Q P = 45 ^ { \circ }\), as shown in Figure 3. The box is modelled as a particle and the ropes are modelled as light inextensible strings. Find

1. the tension in \(B P\),
2. the tension in \(B Q\).

4. \begin{figure}[h] \end{figure} Two identical small rings, \(A\) and \(B\), each of mass \(m\), are threaded onto a rough horizontal wire. The rings are connected by a light inextensible string. A particle \(C\) of mass \(3 m\) is attached to the midpoint of the string. The particle \(C\) hangs in equilibrium below the wire with angle \(B A C = \beta\), as shown in Figure 2. The tension in each of the parts, \(A C\) and \(B C\), of the string is \(T\)

1. By considering particle \(C\), find \(T\) in terms of \(m , g\) and \(\beta\)
2. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between the wire and \(A\). The coefficient of friction between each ring and the wire is \(\frac { 4 } { 5 }\) The two rings, \(A\) and \(B\), are on the point of sliding along the wire towards each other.
3. Find the value of \(\tan \beta\) \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-11_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

3. \begin{figure}[h] \end{figure} A parcel of mass 20 kg is at rest on a rough horizontal floor. The coefficient of friction between the parcel and the floor is 0.3 Two forces, both acting in the same vertical plane, of magnitudes 200 N and \(T \mathrm {~N}\) are applied to the parcel. The line of action of the 200 N force makes an angle of \(15 ^ { \circ }\) with the horizontal and the line of action of the \(T \mathrm {~N}\) force makes an angle of \(25 ^ { \circ }\) with the horizontal, as shown in Figure 1. The parcel is modelled as a particle \(P\). Find the smallest value of \(T\) for which \(P\) remains in equilibrium.

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 5 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a force of magnitude \(F\) newtons. The direction of this force is perpendicular to the string and \(O P\) makes an angle of \(60 ^ { \circ }\) with the vertical, as shown in Figure 1. Find

1. the value of \(F\)
2. the tension in the string.

6. \begin{figure}[h] \end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.

• The tension in the first rope is 500 N acting at an angle of \(40 ^ { \circ }\) to the direction of motion, as shown in Figure 3.
• The tension in the second rope is \(P\) newtons, acting at an angle of \(\alpha ^ { \circ }\) to the direction of motion, also shown in Figure 3.
• The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N

The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find

1. the value of \(\alpha\)
2. the value of \(P\)

1. \begin{figure}[h] \end{figure} Figure 1 shows a small smooth ring threaded onto a light inextensible string.
One end of the string is attached to a fixed point \(A\) on a horizontal ceiling and the other end of the string is attached to a fixed point \(B\) on the ceiling. A horizontal force of magnitude 2 N acts on the ring so that the ring rests in equilibrium at a point \(C\), vertically below \(B\), with the string taut. The line of action of the horizontal force and the string both lie in the same vertical plane. The angle that the string makes with the ceiling at \(A\) is \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\) The tension in the string is \(T\) newtons. The mass of the ring is \(M \mathrm {~kg}\).

1. Find the value of \(T\)
2. Find the value of \(M\)

8. \begin{figure}[h] \end{figure} A fixed rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) A small smooth pulley is fixed at the top of the plane.
One end of a light inextensible string is attached to a particle \(P\) which is at rest on the plane. The string passes over the pulley and the other end of the string is attached to a particle \(Q\) which hangs vertically below the pulley, as shown in Figure 5. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(0.5 m\) The string from \(P\) to the pulley lies along a line of greatest slope of the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\).
The system is in limiting equilibrium with the string taut and \(P\) is on the point of slipping up the plane.

1. Find the value of \(\mu\). The string breaks and \(P\) begins to move down the plane.
   When particle \(P\) has travelled a distance of 0.8 m down the plane, the speed of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the value of \(V\).

3. A particle \(P\) of mass 1.5 kg is placed at a point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.6

1. Show that \(P\) rests in equilibrium at \(A\). A horizontal force of magnitude \(X\) newtons is now applied to \(P\), as shown in Figure 1. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-04_236_584_667_680} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The particle is on the point of moving up the plane.
2. Find
   1. the magnitude of the normal reaction of the plane on \(P\),
   2. the value of \(X\).

1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a particle \(P\).

$$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 3 a \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 2 } = ( 2 a \mathbf { i } + b \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 3 } = ( b \mathbf { i } + 4 \mathbf { j } ) \mathrm { N } .$$ The particle \(P\) is in equilibrium under the action of these forces.
Find the value of \(a\) and the value of \(b\).

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 5 N is attached to one end of a light string. The other end of the string is attached to a fixed point \(O\). A force of magnitude \(F\) newtons is applied to \(P\). The line of action of the force is inclined to the horizontal at \(30 ^ { \circ }\) and lies in the same vertical plane as the string. The particle \(P\) is in equilibrium with the string making an angle of \(40 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Find

1. the tension in the string,
2. the value of \(F\).

5. \begin{figure}[h] \end{figure} A small bead of mass 0.2 kg is attached to the end \(P\) of a light rod \(P Q\). The bead is threaded onto a fixed vertical rough wire. The bead is held in equilibrium with the \(\operatorname { rod } P Q\) inclined to the wire at an angle \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. The thrust in the rod is \(T\) newtons.
The bead is modelled as a particle.

1. Find the magnitude and direction of the friction force acting on the bead when \(T = 2.5\) The coefficient of friction between the bead and the wire is \(\mu\).
   Given that the greatest possible value of \(T\) is 6.125
2. find the value of \(\mu\).

5. \begin{figure}[h] \end{figure} A small ring of mass 0.2 kg is attached to one end of a light inextensible string.
The ring is threaded onto a fixed rough vertical rod.
The string is taut and makes an angle \(\theta\) with the rod, as shown in Figure 3, where \(\tan \theta = \frac { 12 } { 5 }\) Given that the ring is in equilibrium and that the tension in the string is 10 N ,

1. find the magnitude of the frictional force acting on the ring,
2. state the direction of the frictional force acting on the ring. The coefficient of friction between the ring and the rod is \(\frac { 1 } { 4 }\) Given that the ring is in equilibrium, and that the tension in the string, \(T\) newtons, can now vary,
3. 1. find the minimum value of \(T\)
   2. find the maximum value of \(T\)

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 6 kg lies on the surface of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 49 N , acting at an angle \(\theta\) to the plane, as shown in Figure 1. The force acts in a vertical plane through a line of greatest slope of the plane.

1. Show that \(\cos \theta = \frac { 3 } { 5 }\).
2. Find the normal reaction between \(P\) and the plane. The direction of the force of magnitude 49 N is now changed. It is now applied horizontally to \(P\) so that \(P\) moves up the plane. The force again acts in a vertical plane through a line of greatest slope of the plane.
3. Find the initial acceleration of \(P\). \(\_\_\_\_\)}

5. \section*{Figure 2} A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The force acts in a vertical plane containing a line of greatest slope of the plane and has magnitude \(P\) newtons, as shown in Figure 2. The coefficient of friction between the package and the plane is 0.5 and the package is modelled as a particle. The package is in equilibrium and on the point of slipping down the plane.

1. Draw, on Figure 2, all the forces acting on the package, showing their directions clearly.
2. 1. Find the magnitude of the normal reaction between the package and the plane.
   2. Find the value of \(P\).

7. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a block \(P\) of mass 5 kg . The block \(P\) is held at rest on a smooth fixed plane which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The string lies along a line of greatest slope of the plane and passes over a smooth light pulley which is fixed at the top of the plane. The other end of the string is attached to a light scale pan which carries two blocks \(Q\) and \(R\), with block \(Q\) on top of block \(R\), as shown in Figure 3. The mass of block \(Q\) is 5 kg and the mass of block \(R\) is 10 kg . The scale pan hangs at rest and the system is released from rest. By modelling the blocks as particles, ignoring air resistance and assuming the motion is uninterrupted, find

1. 1. the acceleration of the scale pan,
   2. the tension in the string,
2. the magnitude of the force exerted on block \(Q\) by block \(R\),
3. the magnitude of the force exerted on the pulley by the string.

3 Two horizontal forces \(\mathbf { X }\) and \(\mathbf { Y }\) act at a point \(O\) and are at right angles to each other. \(\mathbf { X }\) has magnitude 12 N and acts along a bearing of \(090 ^ { \circ } . \mathbf { Y }\) has magnitude 15 N and acts along a bearing of \(000 ^ { \circ }\).

1. Calculate the magnitude and bearing of the resultant of \(\mathbf { X }\) and \(\mathbf { Y }\).
2. A third force \(\mathbf { E }\) is now applied at \(O\). The three forces \(\mathbf { X } , \mathbf { Y }\) and \(\mathbf { E }\) are in equilibrium. State the magnitude of \(\mathbf { E }\), and give the bearing along which it acts.

2 A trailer of mass 500 kg is attached to a car of mass 1250 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road. The resistance to motion of the trailer is 400 N and the resistance to motion of the car is 900 N . Find both the tension in the tow-bar and the driving force of the car in each of the following cases.

1. The car and trailer are travelling at constant speed.
2. The car and trailer have acceleration \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1
A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m \mathrm {~kg}\) is threaded on the string and is pulled by a force of magnitude 1.6 N acting upwards at \(45 ^ { \circ }\) to the horizontal. The section \(A R\) of the string makes an angle of \(30 ^ { \circ }\) with the downward vertical and the section \(B R\) is horizontal (see diagram). The ring is in equilibrium with the string taut.

1. Give a reason why the tension in the part \(A R\) of the string is the same as that in the part \(B R\).
2. Show that the tension in the string is 0.754 N , correct to 3 significant figures.
3. Find the value of \(m\).

3 A box of mass 5 kg is at rest on a rough horizontal floor.

1. Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of \(40 ^ { \circ }\) to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-2_293_472_2131_794} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Draw a diagram showing all the forces acting on the box.
3. Calculate the new value of the normal reaction of the floor on the box and also the frictional force.

4 A box of mass 2.5 kg is on a smooth horizontal table, as shown in Fig. 4. A light string AB is attached to the table at A and the box at B . AB is at an angle of \(50 ^ { \circ }\) to the vertical. Another light string is attached to the box at C ; this string is inclined at \(15 ^ { \circ }\) above the horizontal and the tension in it is 20 N . The box is in equilibrium. \begin{figure}[h] \end{figure}

1. Calculate the horizontal component of the force exerted on the box by the string at C .
2. Calculate the tension in the string AB .
3. Calculate the normal reaction of the table on the box. The string at C is replaced by one inclined at \(15 ^ { \circ }\) below the horizontal with the same tension of 20 N .
4. Explain why this has no effect on the tension in string AB .

2 Fig. 2 shows a small object, P , of weight 20 N , suspended by two light strings. The strings are tied to points A and B on a sloping ceiling which is at an angle of \(60 ^ { \circ }\) to the upward vertical. The string AP is at \(60 ^ { \circ }\) to the downward vertical and the string BP makes an angle of \(30 ^ { \circ }\) with the ceiling. The object is in equilibrium. \begin{figure}[h] \end{figure}

1. Show that \(\angle \mathrm { APB } = 90 ^ { \circ }\).
2. Draw a labelled triangle of forces to represent the three forces acting on P .
3. Hence, or otherwise, find the tensions in the two strings.

1 Fig. 1 shows a block of mass 3 kg on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.
A force \(P \mathrm {~N}\) is applied to the block parallel to the plane in the upwards direction.
The plane is rough so that a frictional force of 10 N opposes the motion.
The block is moving at constant speed up the plane. \begin{figure}[h] \end{figure}

1. Mark and label all the forces acting on the block.
2. Calculate the magnitude of the normal reaction of the plane on the block.
3. Calculate the magnitude of the force \(P\).

3 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).

1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.

4 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.