Elastic string equilibrium and statics

1 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_113_787_264_680} A light elastic spring of natural length 0.25 m and modulus of elasticity 100 N is held horizontally between two parallel plates. The axis of the spring is at right angles to each of the plates. The horizontal force exerted on the spring by each of the plates is 20 N (see diagram). Find the amount by which the spring is compressed and hence write down the distance between the plates.

1. A light elastic string \(A B\) has natural length \(11 a\) and modulus of elasticity \(6 m g\)

A particle of mass \(4 m\) is attached to the point \(C\) on the string where \(A C = 8 a\) and a particle of mass \(2 m\) is attached to the end \(B\) \begin{figure}[h] \end{figure} The end \(A\) of the string is attached to a fixed point and the string hangs vertically below \(A\) with the particle of mass \(4 m\) in equilibrium at the point \(P\) and the particle of mass \(2 m\) in equilibrium at the point \(Q\), as shown in Figure 1.

1. Find the length \(A P\)
2. Find the length \(P Q\)

2. \begin{figure}[h] \end{figure} A light elastic string AB has natural length I and modulus of elasticity 2 mg .
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 and horizontal, by a force of magnitude F . The line of action of the force and the elastic string lie in the same vertical plane. The direction of the force makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the upward vertical, as shown in Figure 2.
Find, in terms of I , the length AB . \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-04_2264_53_311_1981}

1. \begin{figure}[h] \end{figure} A particle of mass 5 kg is attached to one end of two light elastic strings. The other ends of the strings are attached to a hook on a beam. The particle hangs in equilibrium at a distance 120 cm below the hook with both strings vertical, as shown in Fig. 1. One string has natural length 100 cm and modulus of elasticity 175 N . The other string has natural length 90 cm and modulus of elasticity \(\lambda\) newtons. Find the value of \(\lambda\).
(5)

1. \section*{Figure 1} A particle \(P\) of mass 0.8 kg is attached to one end of a light inelastic string, of natural length 1.2 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(A\). A horizontal force of magnitude \(F\) newtons is applied to \(P\). The particle \(P\) in in equilibrium with the string making an angle \(60 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Calculate

1. the value of \(F\),
2. the extension of the string,
3. the elasticity stored in the string.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a horizontal force of magnitude \(\frac { 4 } { 3 } m g\) applied to \(P\). This force acts in the vertical plane containing the string, as shown in Figure 1. Find (a) the tension in the string,
(b) the elastic energy stored 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 \(O P\).

1. A particle of mass 0.8 kg is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and comes to instantaneous rest 1.1 m below \(A\).

Find the modulus of elasticity of the string.

2. A light elastic string \(A B\) has one end \(A\) attached to a fixed point on a ceiling. A particle \(P\) of mass 0.3 kg is attached to \(B\). When \(P\) hangs in equilibrium with \(A B\) vertical, \(A B = 100 \mathrm {~cm}\). The particle \(P\) is replaced by another particle \(Q\) of mass 0.5 kg . When \(Q\) hangs in equilibrium with \(A B\) vertical, \(A B = 110 \mathrm {~cm}\). Find

1. the natural length of the string,
2. the modulus of elasticity of the string.

1 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A particle of mass 2 kg is attached to one end of a light elastic string of natural length 0.5 metres and modulus of elasticity 100 N . The other end of the string is attached to the point \(O\). Find the extension of the elastic string when the particle hangs in equilibrium vertically below \(O\). Circle your answer.
0.01 m
0.1 m
0.2 m
0.4 m

1 \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-2_435_665_255_699} A small object \(W\) of weight 100 N is attached to one end of each of two parallel light elastic strings. One string is of natural length 0.4 m and has modulus of elasticity 20 N ; the other string is of natural length 0.6 m and has modulus of elasticity 30 N . The upper ends of both strings are attached to a horizontal ceiling and \(W\) hangs in equilibrium at a distance \(d \mathrm {~m}\) below the ceiling (see diagram). Find \(d\).

3 The fixed points A and B lie on a line of greatest slope of a smooth inclined plane, with B higher than A . The horizontal distance from A to B is 2.4 m and the vertical distance is 0.7 m . The fixed point C is 2.5 m vertically above B . A light elastic string of natural length 2.2 m has one end attached to C and the other end attached to a small block of mass 9 kg which is in contact with the plane. The block is in equilibrium when it is at A, as shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Show that the modulus of elasticity of the string is 37.73 N . The block starts at A and is at rest. A constant force of 18 N , acting in the direction AB , is then applied to the block so that it slides along the line AB .
2. Find the magnitude and direction of the acceleration of the block
   (A) when it leaves the point A ,
   (B) when it reaches the point B .
3. Find the speed of the block when it reaches the point B .

2. A particle \(P\) is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point \(A\). \begin{figure}[h] \end{figure} When a horizontal force of magnitude 20 N is applied to \(P\), it rests in equilibrium with the string making an angle of \(30 ^ { \circ }\) with the vertical and \(A P = 1.2 \mathrm {~m}\) as shown in Figure 1.

1. Find the tension in the string.
2. Find the elastic potential energy stored in the string.

1 A small object of mass 5 kg is attached to one end of each of two identical parallel light elastic strings. The upper ends of both strings are attached to a horizontal ceiling.
The object hangs in equilibrium at R , with the extension of each string being 0.1 m , as shown in Fig. 1. \begin{figure}[h] \end{figure}

1. Find the stiffness of each string. One of the strings is now removed and the object initially falls downwards. The object does not return to R at any point in the subsequent motion.
2. Suggest a reason why the object does not return to \(R\).

1 The end O of a light elastic string OA is attached to a fixed point.
Fiona attaches a mass of 1 kg to the string at A . The system hangs vertically in equilibrium and the length of the stretched string is 70 cm . Fiona removes the 1 kg mass and attaches a mass of 2 kg to the string at A . The system hangs vertically in equilibrium and the length of the stretched string is now 80 cm . Fiona then removes the 2 kg mass and attaches a mass of 5 kg to the string at A . The system hangs vertically in equilibrium.

1. Use the information given in the question to determine expected values for
   Fiona discovers that, when the mass of 5 kg is attached to the string at A , the length of the stretched string is greater than the expected length.
2. Suggest a reason why this has happened.

3 One end of a light elastic spring of natural length 0.3 m is attached to a fixed point. A mass of 4 kg is attached to the other end of the spring. When the spring hangs vertically in equilibrium the extension of the spring is 0.02 m .

1. Determine the modulus of elasticity of the spring. A student calculates that if the mass of 4 kg is removed and replaced with a mass of 20 kg the extension of the spring will be 0.1 m .
2. Suggest a reason why this extension may not be 0.1 m .

\includegraphics{figure_4} A uniform rod \(AB\) of length \(3a\) and weight \(W\) is freely hinged to a fixed point at the end \(A\). The end \(B\) is below the level of \(A\) and is attached to one end of a light elastic string of natural length \(4a\). The other end of the string is attached to a point \(O\) on a vertical wall. The horizontal distance between \(A\) and the wall is \(5a\). The string and the rod make angles \(\theta\) and \(2\theta\) respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that \(\sin \theta = \frac{3}{5}\) and you may use the fact that \(\cos 2\theta = \frac{7}{25}\).

1. Find the tension in the string in terms of \(W\). [3]
2. Find the modulus of elasticity of the string in terms of \(W\). [4]
3. Find the angle that the force acting on the rod at \(A\) makes with the horizontal. [3]

\includegraphics{figure_1} A particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\) on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the equilibrium position, the string makes an angle \(\theta\) with the ceiling (see diagram). The string has natural length 0.8 m and modulus of elasticity 50 N.

1. Find the tension in the string. [2]
2. Find the vertical distance between the particle and the ceiling. [3]

A thin elastic string, of modulus \(\lambda\) N and natural length 20 cm, passes round two small, smooth pegs \(A\) and \(B\) on the same horizontal level to form a closed loop. \(AB = 10\) cm. The ends of the string are attached to a weight \(P\) of mass 0.7 kg. When \(P\) rests in equilibrium, \(APB\) forms an equilateral triangle. \includegraphics{figure_2}

1. Find the value of \(\lambda\). [6 marks]
2. State one assumption that you have made about the weight \(P\), explaining how you have used this assumption in your solution. [1 mark]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length 0·5 m and modulus of elasticity \(\frac{mg}{2}\) N. The other end of the string is attached to a fixed point \(O\) and \(P\) hangs vertically below \(O\).

1. Find the stretched length of the string when \(P\) rests in equilibrium. [3 marks]
2. Find the elastic potential energy stored in the string in the equilibrium position. [2 marks]

\(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.

1. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position. [2 marks]
2. Explain the discrepancy between your answers to parts (b) and (c). [1 mark]

A particle \(P\) of mass 0.2 kg moves in a horizontal circle on one end of an elastic string whose other end is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\pi\) rad s\(^{-1}\). The natural length of the string is 1 m and, while \(P\) is in motion, the distance \(OP = 1.15\) m.

1. Calculate, to 3 significant figures, the modulus of elasticity of the string. [6 marks]

The motion now ceases and \(P\) hangs at rest vertically below \(O\).

1. Show that the extension in the string in this position is about 13 cm. [3 marks]

\(ABCD\) is a uniform lamina in the shape of a kite with \(BA = BC = 0.37\) m, \(DA = DC = 0.91\) m and \(AC = 0.7\) m (see diagram). The centre of mass of \(ABCD\) is \(G\). \includegraphics{figure_4}

1. Explain why \(G\) lies on \(BD\). [1]
2. Show that the distance of \(G\) from \(B\) is \(0.36\) m. [4]

The lamina \(ABCD\) is freely suspended from the point \(A\).

1. Determine the acute angle that \(CD\) makes with the horizontal, stating which of \(C\) or \(D\) is higher. [4]