3.03n Equilibrium in 2D: particle under forces

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.

1. Find the size of the angle \(\alpha\).
2. Find the 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 \(P\).

6. Two forces, \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\), act on a particle \(P\) of mass \(m \mathrm {~kg}\). The resultant of the two forces is \(\mathbf { R }\). Given that \(\mathbf { R }\) acts in a direction which is parallel to the vector ( \(\mathbf { i } - 2 \mathbf { j }\) ),

1. find the angle between \(\mathbf { R }\) and the vector \(\mathbf { j }\),
2. show that \(2 p + q + 3 = 0\). Given also that \(q = 1\) and that \(P\) moves with an acceleration of magnitude \(8 \sqrt { } 5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), (c) find the value of \(m\).

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 \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_437_846_274_651} A block of mass 50 kg is in equilibrium on smooth horizontal ground with one end of a light wire attached to its upper surface. The other end of the wire is attached to an object of mass mkg . The wire passes over a small smooth pulley, and the object hangs vertically below the pulley. The part of the wire between the block and the pulley makes an angle of \(72 ^ { \circ }\) with the horizontal. A horizontal force of magnitude X N acts on the block in the vertical plane containing the wire (see diagram). The tension in the wire is T N and the contact force exerted by the ground on the block is R N.

1. By resolving forces on the block vertically, find a relationship between T and R . It is given that the block is on the point of lifting off the ground.
2. Show that \(\mathrm { T } = 515\), correct to 3 significant figures, and hence find the value of m .
3. By resolving forces on the block horizontally, write down a relationship between T and X , and hence find the value of \(X\).

3 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.

1. Find the values of \(x , y\) and \(z\).
2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).

5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on the box.
2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.

2 Fig. 2 shows a light string with an object of mass 4 kg attached at end A . The string passes over a smooth pulley and its other end B is attached to two light strings BC and BD of the same length. The strings BC and BD are attached to horizontal ground and are each inclined at \(20 ^ { \circ }\) to the vertical. The system is in equilibrium. \begin{figure}[h] \end{figure}

1. What information in the question tells you that the tension is the same throughout the string AB ?
2. What is the tension in the string AB ?
3. Calculate the tension in the strings BC and BD .

3 Fig. 3 shows a smooth ball resting in a rack. The angle in the middle of the rack is \(90 ^ { \circ }\). The rack has one edge at angle \(\alpha\) to the horizontal. The weight of the ball is \(W \mathrm {~N}\). The reaction forces of the rack on the ball at the points of contact are \(R \mathrm {~N}\) and \(S \mathrm {~N}\). \begin{figure}[h] \end{figure}

1. Draw a fully labelled triangle of forces to show the forces acting on the ball. Your diagram must indicate which angle is \(\alpha\).
2. Find the values of \(R\) and \(S\) in terms of \(W\) and \(\alpha\).
3. On the same axes draw sketches of \(R\) against \(\alpha\) and \(S\) against \(\alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 90 ^ { \circ }\). For what values of \(\alpha\) is \(R < S\) ?

1 Fig. 1 shows four forces acting at a point. The forces are in equilibrium. \begin{figure}[h] \end{figure} Show that \(P = 14\). Find \(Q\), giving your answer correct to 3 significant figures.

3 Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h] \end{figure} Abi pulls vertically downwards on the rope A with a force \(F\) N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle \(\theta\) with the vertical.

1. Represent the forces acting on the object as a fully labelled triangle of forces.
2. Find \(F\) and \(\theta\). Show that the distance between the object and the vertical section of rope A is 3 m . Abi then pulls harder and the object moves upwards. Bob adjusts the tension in rope B so that the object moves along a vertical line. Fig. 7.2 shows the situation when the object is part of the way up. The tension in rope A is \(S \mathrm {~N}\) and the tension in rope B is \(T \mathrm {~N}\). The ropes make angles \(\alpha\) and \(\beta\) with the vertical as shown in the diagram. Abi and Bob are taking a rest and holding the object stationary and in equilibrium. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{82f933a6-c17e-4b41-ae2b-3cc9d0ba975c-3_384_357_520_851} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Give the equations, involving \(S , T , \alpha\) and \(\beta\), for equilibrium in the vertical and horizontal directions.
4. Find the values of \(S\) and \(T\) when \(\alpha = 8.5 ^ { \circ }\) and \(\beta = 35 ^ { \circ }\).
5. Abi's mass is 40 kg . Explain why it is not possible for her to raise the object to a position in which \(\alpha = 60 ^ { \circ }\).

6 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string \(B C\) is fixed at \(C\). The end \(A\) of string \(A B\) is fixed so that \(A B\) is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on the box.
2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.

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.

2. \begin{figure}[h] \end{figure} A light elastic string \(A B\) has modulus of elasticity \(2 m g\) and natural length \(k a\), where \(k\) is a constant.
The end \(A\) of the elastic string is attached to a fixed point. The other end \(B\) is attached to a particle of mass \(m\). The particle is held in equilibrium, with the elastic string taut, by a force that acts in a direction that is perpendicular to the string. The line of action of the force and the elastic string lie in the same vertical plane. The string makes an angle \(\theta\) with the downward vertical at \(A\), as shown in Figure 2. Given that the length \(A B = \frac { 21 } { 10 } a\) and \(\tan \theta = \frac { 3 } { 4 }\), find the value of \(k\).

1. \begin{figure}[h] \end{figure} A light elastic string \(A B\) has natural length \(4 a\) and modulus of elasticity \(\lambda\). The end \(A\) is attached to a fixed point and the end \(B\) is attached to a particle of mass \(m\). The particle is held in equilibrium, with the string stretched, by a horizontal force of magnitude \(k m g\).
The line of action of the horizontal force lies in the vertical plane containing the elastic string.
The string \(A B\) makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac { 4 } { 3 }\) With the particle in this position, \(A B = 5 a\), as shown in Figure 1.

1. Show that \(\lambda = \frac { 20 m g } { 3 }\)
2. Find the value of \(k\).

5. \begin{figure}[h] \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded on to a thin smooth fixed vertical pole. One end of a light inextensible string of length \(2 l\) is attached to a point \(A\) on the pole. The other end of the string is attached to \(R\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle \(P\) moves with constant angular speed in a horizontal circle, with both halves of the string taut, and \(A R = \frac { 6 l } { 5 }\), as shown in Figure 2. It may be assumed that in this motion the string does not wrap itself around the pole and that at any instant, the triangle \(A P R\) lies in a vertical plane.

1. Show that the tension in the lower half of the string is \(\frac { 5 m g } { 3 }\)
2. Find, in terms of \(l\) and \(g\), the time for \(P\) to complete one revolution.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is fixed to a point \(A\) which is vertically above the point \(O\) on a smooth horizontal table. The particle \(P\) remains in contact with the surface of the table and moves in a circle with centre \(O\) and with angular speed \(\sqrt { \left( \frac { k g } { 3 a } \right) }\), where \(k\) is a constant. Throughout the motion the string remains taut and \(\angle A P O = 30 ^ { \circ }\), as shown in Figure 3.

1. Show that the tension in the string is \(\frac { 2 k m g } { 3 }\).
2. Find, in terms of \(m , g\) and \(k\), the normal reaction between \(P\) and the table.
3. Deduce the range of possible values of \(k\). The angular speed of \(P\) is changed to \(\sqrt { \left( \frac { 2 g } { a } \right) }\). The particle \(P\) now moves in a horizontal circle above the table. The centre of this circle is \(X\).
4. Show that \(X\) is the mid-point of \(O A\).

5. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.

1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
3. find, in terms of \(m\) and \(g\), the tension in the string.

4. \begin{figure}[h] \end{figure} A particle \(P\) of weight 40 N is attached to one end of a light elastic string of natural length 0.5 m . The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 30 N is applied to \(P\), as shown in Figure 3. The particle \(P\) is in equilibrium and the elastic energy stored in the string is 10 J . Calculate the length \(O P\).

4. \begin{figure}[h] \end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves in a horizontal circle of radius \(3 a\) with angular speed \(\omega\). The centre of the circle is \(C\) where \(C\) lies on \(A B\) and \(A C = 4 a\), as shown in Figure 4. Both parts of the string are taut.

1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 3 a \omega ^ { 2 } + g \right)\).
2. Find the tension in \(B P\).
3. Deduce that \(\omega \geqslant \frac { 1 } { 2 } \sqrt { } \left( \frac { g } { a } \right)\).

4 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_494_255_258_945} Particles \(P\) and \(Q\), of masses 0.4 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley and the sections of the string not in contact with the pulley are vertical. \(P\) rests in limiting equilibrium on a plane inclined at \(60 ^ { \circ }\) to the horizontal (see diagram).

1. (a) Calculate the components, parallel and perpendicular to the plane, of the contact force exerted by the plane on \(P\).
   (b) Find the coefficient of friction between \(P\) and the plane. \(P\) is held stationary and a particle of mass 0.2 kg is attached to \(Q\). With the string taut, \(P\) is released from rest.
2. Calculate the tension in the string and the acceleration of the particles. \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_579_1195_1553_475} The \(( t , v )\) diagram represents the motion of two cyclists \(A\) and \(B\) who are travelling along a horizontal straight road. At time \(t = 0 , A\), who cycles with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), overtakes \(B\) who has initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). From time \(t = 0 B\) cycles with constant acceleration for 20 s . When \(t = 20\) her speed is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), which she subsequently maintains.

6 \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_348_1109_1345_516} A small smooth ring \(R\) of weight 7 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\) at the same horizontal level. A horizontal force of magnitude 5 N is applied to \(R\). The string is taut. In the equilibrium position the angle \(A R B\) is a right angle, and the portion of the string attached to \(B\) makes an angle \(\theta\) with the horizontal (see diagram).

1. Explain why the tension \(T \mathrm {~N}\) is the same in each part of the string.
2. By resolving horizontally and vertically for the forces acting on \(R\), form two simultaneous equations in \(T \cos \theta\) and \(T \sin \theta\).
3. Hence find \(T\) and \(\theta\).

6 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_328_698_255_657} A particle \(P\) lies on a slope inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is attached to one end of a taut light inextensible string which passes through a small smooth ring \(Q\) of mass \(m \mathrm {~kg}\). The portion \(P Q\) of the string is horizontal and the other portion of the string is inclined at \(40 ^ { \circ }\) to the vertical. A horizontal force of magnitude \(H \mathrm {~N}\), acting away from \(P\), is applied to \(Q\) (see diagram). The tension in the string is 6.4 N , and the string is in the vertical plane containing the line of greatest slope on which \(P\) lies. Both \(P\) and \(Q\) are in equilibrium.

1. Calculate \(m\).
2. Calculate \(H\).
3. Given that the weight of \(P\) is 32 N , and that \(P\) is in limiting equilibrium, show that the coefficient of friction between \(P\) and the slope is 0.879 , correct to 3 significant figures. \(Q\) and the string are now removed.
4. Determine whether \(P\) remains in equilibrium.

4 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).

1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
2. the magnitude of the force exerted on \(P\) by the surface,
3. the angle between the surface and the 10 N force.