Particle suspended by strings

3
\includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-05_518_616_255_767} A particle \(Q\) of mass 0.2 kg is held in equilibrium by two light inextensible strings \(P Q\) and \(Q R . P\) is a fixed point on a vertical wall and \(R\) is a fixed point on a horizontal floor. The angles which strings \(P Q\) and \(Q R\) make with the horizontal are \(60 ^ { \circ }\) and \(30 ^ { \circ }\) respectively (see diagram). Find the tensions in the two strings.

3
\includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-05_446_851_260_646} A block of mass \(m \mathrm {~kg}\) is held in equilibrium below a horizontal ceiling by two strings, as shown in the diagram. One of the strings is inclined at \(45 ^ { \circ }\) to the horizontal and the tension in this string is \(T \mathrm {~N}\). The other string is inclined at \(60 ^ { \circ }\) to the horizontal and the tension in this string is 20 N . Find \(T\) and \(m\).

3
\includegraphics[max width=\textwidth, alt={}, center]{172e83d8-730c-4c18-aacb-ba2596886e41-04_412_601_260_772} A particle of mass 0.3 kg is held at rest by two light inextensible strings. One string is attached at an angle of \(60 ^ { \circ }\) to a horizontal ceiling. The other string is attached at an angle \(\alpha ^ { \circ }\) to a vertical wall (see diagram). The tension in the string attached to the ceiling is 4 N . Find the tension in the string which is attached to the wall and find the value of \(\alpha\).

2
\includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-03_280_588_264_774} A particle of mass 2.4 kg is held in equilibrium by two light inextensible strings, one of which is attached to point \(A\) and the other attached to point \(B\). The strings make angles of \(35 ^ { \circ }\) and \(40 ^ { \circ }\) with the horizontal (see diagram). Find the tension in each of the two strings.

4
\includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_389_1134_258_468} The diagram shows two particles, \(A\) and \(B\), of masses 0.2 kg and 0.1 kg respectively. The particles are suspended below a horizontal ceiling by two strings, \(A P\) and \(B Q\), attached to fixed points \(P\) and \(Q\) on the ceiling. The particles are connected by a horizontal string, \(A B\). Angle \(A P Q = 45 ^ { \circ }\) and \(B Q P = \theta ^ { \circ }\). Each string is light and inextensible. The particles are in equilibrium.

1. Find the value of the tension in the string \(A B\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_2715_44_110_2006}
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-07_2721_34_101_20}
2. Find the value of \(\theta\) and the tension in the string \(B Q\).

3
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-2_479_771_1356_687} Each of three light strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(A\). The strings are in equilibrium with two of them passing over fixed smooth horizontal pegs, and with the particles hanging freely. The weights of the particles, and the angles between the sloping parts of the strings and the vertical, are as shown in the diagram. Find the values of \(W _ { 1 }\) and \(W _ { 2 }\).
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3
\includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_606_843_1731_651} The diagram shows three particles \(A , B\) and \(C\) hanging freely in equilibrium, each being attached to the end of a string. The other ends of the three strings are tied together and are at the point \(X\). The strings carrying \(A\) and \(C\) pass over smooth fixed horizontal pegs \(P _ { 1 }\) and \(P _ { 2 }\) respectively. The weights of \(A , B\) and \(C\) are \(5.5 \mathrm {~N} , 7.3 \mathrm {~N}\) and \(W \mathrm {~N}\) respectively, and the angle \(P _ { 1 } X P _ { 2 }\) is a right angle. Find the angle \(A P _ { 1 } X\) and the value of \(W\).

2
\includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_342_629_616_756} The diagram shows a small object \(P\) of mass 20 kg held in equilibrium by light ropes attached to fixed points \(A\) and \(B\). The rope \(P A\) is inclined at an angle of \(50 ^ { \circ }\) above the horizontal, the rope \(P B\) is inclined at an angle of \(10 ^ { \circ }\) below the horizontal, and both ropes are in the same vertical plane. Find the tension in the rope \(P A\) and the tension in the rope \(P B\).

1. \begin{figure}[h] \end{figure} A particle of weight \(W\) newtons is attached at \(C\) to the ends of two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively, as shown in Fig.1. Given the tension in \(A C\) is 50 N , calculate

1. the tension in \(B C\), to 3 significant figures,
2. the value of \(W\).

1. \begin{figure}[h] \end{figure} A particle of weight \(W\) newtons is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(50 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(B C\) is 6 N , find

1. the tension in \(A C\),
2. the value of \(W\).

3. \begin{figure}[h] \end{figure} A particle of mass 2 kg is suspended from a horizontal ceiling by two light inextensible strings, \(P R\) and \(Q R\). The particle hangs at \(R\) in equilibrium, with the strings in a vertical plane. The string \(P R\) is inclined at \(55 ^ { \circ }\) to the horizontal and the string \(Q R\) is inclined at \(35 ^ { \circ }\) to the horizontal, as shown in Figure 1. \section*{Find}

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

1. \begin{figure}[h] \end{figure} A particle of weight \(W\) is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with the strings in a vertical plane and with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(45 ^ { \circ }\) respectively, as shown in Figure 1. Find, in terms of \(W\),

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

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 \(T\).

1. \begin{figure}[h] \end{figure} A particle of weight 24 N is held in equilibrium by two light inextensible strings. One string is horizontal. The other string is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 1. The tension in the horizontal string is \(Q\) newtons and the tension in the other string is \(P\) newtons. Find

1. the value of \(P\),
2. the value of \(Q\).

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.

7 A box of weight 147 N is held by light strings AB and BC . As shown in Fig. 7.1, AB is inclined at \(\alpha\) to the horizontal and is fixed at A ; BC is held at C . The box is in equilibrium with BC horizontal and \(\alpha\) such that \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). \begin{figure}[h] \end{figure}

1. Calculate the tension in string AB .
2. Show that the tension in string BC is 196 N . As shown in Fig. 7.2, a box of weight 90 N is now attached at C and another light string CD is held at D so that the system is in equilibrium with BC still horizontal. CD is inclined at \(\beta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-5_387_702_1343_646} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Explain why the tension in the string BC is still 196 N .
4. Draw a diagram showing the forces acting on the box at C . Find the angle \(\beta\) and show that the tension in CD is 216 N , correct to three significant figures. The string section CD is now taken over a smooth pulley and attached to a block of mass \(M \mathrm {~kg}\) on a rough slope inclined at \(40 ^ { \circ }\) to the horizontal. As shown in Fig. 7.3, the part of the string attached to the box is still at \(\beta\) to the horizontal and the part attached to the block is parallel to the slope. The system is in equilibrium with a frictional force of 20 N acting on the block up the slope. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-6_430_1045_493_502} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
   \end{figure}
5. Calculate the value of \(M\).

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.

3. A particle \(P\) of mass 2 kg is attached to one end of a light string, the other end of which is attached to a fixed point \(O\). The particle is held in equilibrium, with \(O P\) at \(30 ^ { \circ }\) to the downward vertical, by a force of magnitude \(F\) newtons. The force acts in the same vertical plane as the string and acts at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 3. \begin{figure}[h] \end{figure} Find

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

13 A 15 kg box is suspended in the air by a rope which makes an angle of \(30 ^ { \circ }\) with the vertical. The box is held in place by a string which is horizontal.

1. Draw a diagram showing the forces acting on the box.
2. Calculate the tension in the rope.
3. Calculate the tension in the string.

3 A particle hangs at the end of a string. A horizontal force of magnitude \(F \mathrm {~N}\) acting on the particle holds it in equilibrium so that the string makes an angle of \(20 ^ { \circ }\) with the vertical, as shown in the diagram. The tension in the string is 12 N .
\includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-04_357_374_1409_239}

1. Find the value of \(F\).
2. Find the mass of the particle.

3 A particle, of mass 4 kg , is suspended in equilibrium by two light strings, \(A P\) and \(B P\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) to the horizontal and the other string, \(B P\), is horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-2_231_757_1841_639}

1. Draw and label a diagram to show the forces acting on the particle.
2. Show that the tension in the string \(A P\) is 78.4 N .
3. Find the tension in the horizontal string \(B P\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a cable car \(C\) of mass 1 tonne which has broken down. The cable car is suspended in equilibrium by two perpendicular cables \(A C\) and \(B C\) which are attached to fixed points \(A\) and \(B\), at the same horizontal level on either side of a valley. The cable \(A C\) is inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac { 3 } { 4 }\).

1. Show that the tension in the cable \(A C\) is 5880 N and find the tension in the cable BC. A gust of wind then blows along the valley.
2. Explain the effect that this will have on the tension in the two cables.

6 A box of weight 147 N is held by light strings AB and BC . As shown in Fig. 7.1, AB is inclined at \(\alpha\) to the horizontal and is fixed at \(\mathrm { A } ; \mathrm { BC }\) is held at C . The box is in equilibrium with BC horizontal and \(\alpha\) such that \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). \begin{figure}[h] \end{figure}

1. Calculate the tension in string AB .
2. Show that the tension in string BC is 196 N . As shown in Fig. 7.2, a box of weight 90 N is now attached at C and another light string CD is held at D so that the system is in equilibrium with BC still horizontal. CD is inclined at \(\beta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-4_378_695_1282_687} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Explain why the tension in the string BC is still 196 N .
4. Draw a diagram showing the forces acting on the box at C . Find the angle \(\beta\) and show that the tension in CD is 216 N , correct to three significant figures. The string section CD is now taken over a smooth pulley and attached to a block of mass \(M \mathrm {~kg}\) on a rough slope inclined at \(40 ^ { \circ }\) to the horizontal. As shown in Fig. 7.3, the part of the string attached to the box is still at \(\beta\) to the horizontal and the part attached to the block is parallel to the slope. The system is in equilibrium with a frictional force of 20 N acting on the block up the slope. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-5_436_1049_524_536} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
   \end{figure}
5. Calculate the value of \(M\).