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 \(m\).

6 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-08_412_588_260_776} A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal force of magnitude \(X \mathrm {~N}\), as shown in the diagram. The two strings are both inclined at \(60 ^ { \circ }\) to the vertical.

1. Given that \(X = 100\), find the tension in the lower string.
2. Find the least value of \(X\) for which the block remains in equilibrium in the position shown. [4]

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 \(B Q\).

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 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 \(T\).

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.

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.

5 Two ropes are attached to a load of mass 500 kg . The ropes make angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) to the vertical, as shown in the diagram. The tensions in these ropes are \(T _ { 1 }\) and \(T _ { 2 }\) newtons. The load is also supported by a vertical spring. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_533_565_507_744} The system is in equilibrium and \(T _ { 1 } = 200\).

1. Show that \(T _ { 2 } = 141\), correct to three significant figures.
2. Find the force that the spring exerts on the load.

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 \(B P\).

2. \begin{figure}[h] \end{figure} Figure 2 shows a washing line suspended at either end by vertical rigid poles. A jacket of mass 0.7 kg is suspended in equilibrium part of the way along the line. The sections of the washing line on either side of the jacket make angles of \(35 ^ { \circ }\) and \(40 ^ { \circ }\) with the horizontal.

1. Find the tension in the washing line on each side of the jacket.
2. Explain why, in practice, the angles are likely to be very similar in value.

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 \(M\).

4 A small box has weight \(\mathbf { W } \mathrm { N }\) and is held in equilibrium by two strings with tensions \(\mathbf { T } _ { 1 } \mathrm {~N}\) and \(\mathbf { T } _ { 2 } \mathrm {~N}\). This situation is shown in Fig. 2 which also shows the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) that are horizontal and vertically upwards, respectively. \begin{figure}[h] \end{figure} The tension \(\mathbf { T } _ { 1 }\) is \(10 \mathbf { i } + 24 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { T } _ { 1 }\) and the angle between \(\mathbf { T } _ { 1 }\) and the vertical. The magnitude of the weight is \(w \mathrm {~N}\).
2. Write down the vector \(\mathbf { W }\) in terms of \(w\) and \(\mathbf { j }\). The tension \(\mathbf { T } _ { 2 }\) is \(k \mathbf { i } + 10 \mathbf { j }\), where \(k\) is a scalar.
3. Find the values of \(k\) and of \(w\).

3 A sign, of mass 2 kg , is suspended from the ceiling of a supermarket by two light strings. It hangs in equilibrium with each string making an angle of \(35 ^ { \circ }\) to the vertical, as shown in the diagram. Model the sign as a particle. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-2_424_385_1790_824}

1. By resolving forces horizontally, show that the tension is the same in each string.
2. Find the tension in each string.
3. If the tension in a string exceeds 40 N , the string will break. Find the mass of the heaviest sign that could be suspended as shown in the diagram.

18 An object, \(O\), of mass \(m\) kilograms is hanging from a ceiling by two light, inelastic strings of different lengths. The shorter string, of length 0.8 metres, is fixed to the ceiling at \(A\).
The longer string, of length 1.2 metres, is fixed to the ceiling at \(B\).
This object hangs 0.6 metres directly below the ceiling as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-28_252_940_667_552} 18

1. Show that the tension in the shorter string is over \(30 \%\) more than the tension in the longer string.
   18
2. The tension in the longer string is known to be \(2 g\) newtons. Find the value of \(m\).
   A rough wooden ramp is 10 metres long and is inclined at an angle of \(25 ^ { \circ }\) above the horizontal. The bottom of the ramp is at the point \(O\). A crate of mass 20 kg is at rest at the point \(A\) on the ramp.
   The crate is pulled up the ramp using a rope attached to the crate.
   Once in motion, the rope remains taut and parallel to the line of greatest slope of the ramp. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-30_252_842_804_598}

11 A decoration is hanging freely from a fixed point on a ceiling.
The decoration has a mass of 0.2 kilograms.
The decoration is hanging by a light, inextensible wire.
The wire is 0.1 metres long.
Find the tension in the wire. Circle your answer.
0.02 N
0.02 g N
0.2 N
0.2 g N

\includegraphics{figure_3} A particle of mass 0.3 kg is held at rest by two light inextensible strings. One string is attached at an angle of \(60°\) to a horizontal ceiling. The other string is attached at an angle \(α°\) 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 \(α\). [6]