Particle at midpoint of string between two horizontal fixed points: vertical motion

7 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-4_232_905_762_621} A light elastic string has natural length 10 m and modulus of elasticity 130 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. A small stone is attached to the mid-point of the string and hangs in equilibrium at a point 2.5 m below \(A B\), as shown in the diagram. With the stone in this position the length of the string is 13 m .

1. Find the tension in the string.
2. Show that the mass of the stone is 3 kg . The stone is now held at rest at a point 8 m vertically below the mid-point of \(A B\).
3. Find the elastic potential energy of the string in this position.
4. The stone is now released. Find the speed with which it passes through the mid-point of \(A B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-4_324_1267_794_440} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(A B\) (see diagram).

1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N . \(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
2. Find the speed with which \(P\) passes through \(M\).

4 The ends of a light elastic string of natural length 0.8 m and modulus of elasticity \(\lambda \mathrm { N }\) are attached to fixed points \(A\) and \(B\) which are 1.2 m apart at the same horizontal level. A particle of mass 0.3 kg is attached to the centre of the string, and released from rest at the mid-point of \(A B\). The particle descends 0.32 m vertically before coming to instantaneous rest.

1. Calculate \(\lambda\).
2. Calculate the speed of the particle when it is 0.25 m below \(A B\).

4 A light elastic string has natural length 2.4 m and modulus of elasticity 21 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is projected vertically upwards with velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the mid-point of \(A B\). In the subsequent motion \(P\) is at instantaneous rest at a point 1.6 m above \(A B\).

1. Find \(m\).
2. Calculate the acceleration of \(P\) when it first passes through a point 0.5 m below \(A B\).

3 A light elastic string has natural length 2.2 m and modulus of elasticity 14.3 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is released from rest at the mid-point of \(A B\). In the subsequent motion \(P\) has its greatest speed at a point 0.5 m below \(A B\).

1. Find \(m\).
2. Calculate the greatest speed of \(P\).

5 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_577_693_1740_724} A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light elastic string of natural length 5.5 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 6 m apart. \(P\) is held at rest at a point 1.25 m vertically above the mid-point of \(A B\) and then released. \(P\) travels a distance 5.25 m downwards before coming to instantaneous rest (see diagram). By considering the changes in gravitational potential energy and elastic potential energy as \(P\) travels downwards, find the value of \(\lambda\).

5 A particle \(P\) of mass 0.28 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 4.8 m apart. \(P\) is released from rest at the mid-point of \(A B\). In the subsequent motion, the acceleration of \(P\) is zero when \(P\) is at a distance 0.7 m below \(A B\).

1. Show that the modulus of elasticity of the string is 20 N .
2. Calculate the maximum speed of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-06_264_839_260_653} A light elastic string has natural length 2 m and modulus of elasticity 39 N . The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 2.4 m apart. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below \(A B\) (see diagram).

1. Show that \(m = 0.9\). \(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 1.6 m below \(A B\).
2. Calculate the speed of projection of \(P\).

4. Fixed points \(A\) and \(B\) are on a horizontal ceiling, where \(A B = 4 a\). A light elastic string has natural length \(3 a\) and modulus of elasticity \(\lambda\). One end of the string is attached to \(A\) and the other end is attached to \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at the point \(C\), where \(C\) is at a distance \(\frac { 3 } { 2 } a\) vertically below the ceiling.

1. Show that \(\lambda = \frac { 5 m g } { 4 }\) (5) The point \(D\) is the midpoint of \(A B\). The particle is now raised vertically upwards to \(D\), and released from rest.
2. Find the speed of \(P\) as it passes through \(C\).
   

   \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-05_542_51_2026_1982}

   VIIV SIHI NI JIIIM IONOO

   VI4V SIHI NI JIIYM ION OO

4. \begin{figure}[h] \end{figure} The ends of a light elastic string, of natural length \(4 l\) and modulus of elasticity \(\lambda\), are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 4 l\). A particle \(P\) of mass \(2 m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at a distance \(\frac { 3 } { 2 } l\) vertically below the midpoint of \(A B\), as shown in Figure 2.

1. Show that \(\lambda = \frac { 20 } { 3 } m g\). The particle is pulled vertically downwards from its equilibrium position until the total length of the string is 6l. The particle is then released from rest.
2. Show that \(P\) comes to instantaneous rest before reaching the line \(A B\).

1. A particle \(P\) of mass 1.2 kg is attached to the midpoint of a light elastic string of natural length 0.5 m and modulus of elasticity \(\lambda\) newtons.

The fixed points \(A\) and \(B\) are 0.8 m apart on a horizontal ceiling. One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). Initially \(P\) is held at rest at the midpoint \(M\) of the line \(A B\) and the tension in the string is 30 N .

1. Show that \(\lambda = 50\) The particle is now held at rest at the point \(C\), where \(C\) is 0.3 m vertically below \(M\). The particle is released from rest.
2. Find the magnitude of the initial acceleration of \(P\)
3. Find the speed of \(P\) at the instant immediately before it hits the ceiling.

2. \begin{figure}[h] \end{figure} A smooth bead of weight 12 N is threaded onto a light elastic string of natural length 3 m . The points \(A\) and \(B\) are on a horizontal ceiling, with \(A B = 3 \mathrm {~m}\). One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). The bead hangs freely in equilibrium, 2 m below the ceiling, as shown in Figure 2.

1. Find the tension in the string.
2. Show that the modulus of elasticity of the string is 11.25 N . The bead is now pulled down to a point vertically below its equilibrium position and released from rest.
3. Find the elastic energy stored in the string at the instant when the bead is moving at its maximum speed.

7. A particle \(P\) of mass 1.5 kg is attached to the mid-point of a light elastic string of natural length 0.30 m and modulus of elasticity \(\lambda\) newtons. The ends of the string are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 0.48 \mathrm {~m}\). Initially \(P\) is held at rest at the mid-point, \(M\), of the line \(A B\) and the tension in the string is 240 N .

1. Show that \(\lambda = 400\) The particle is now held at rest at the point \(C\), where \(C\) is 0.07 m vertically below \(M\). The particle is released from rest at \(C\).
2. Find the magnitude of the initial acceleration of \(P\).
3. Find the speed of \(P\) as it passes through \(M\).

2. \begin{figure}[h] \end{figure} Two light elastic strings each have natural length \(a\) and modulus of elasticity \(\lambda\). A particle \(P\) of mass \(m\) is attached to one end of each string. The other ends of the strings are attached to points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 2 a\). The particle is held at the mid-point of \(A B\) and released from rest. It comes to rest for the first time in the subsequent motion when \(P A\) and \(P B\) make angles \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Fig. 1. Find \(\lambda\) in terms of \(m\) and \(g\).

2. \begin{figure}[h] \end{figure} Two elastic ropes each have natural length 30 cm and modulus of elasticity \(\lambda \mathrm { N }\). One end of each rope is attached to a lead weight \(P\) of mass 2 kg and the other ends are attached to two points \(A\) and \(B\) on a horizontal ceiling, where \(A B = 72 \mathrm {~cm}\). The weight hangs in equilibrium 15 cm below the ceiling, as shown in Fig. 2. By modelling \(P\) as a particle and the ropes as light elastic strings,

1. find, to one decimal place, the value of \(\lambda\).
2. State how you have used the fact that \(P\) is modelled as a particle.

3 Two fixed points X and Y are 14.4 m apart and XY is horizontal. The midpoint of XY is M . A particle P is connected to X and to Y by two light elastic strings. Each string has natural length 6.4 m and modulus of elasticity 728 N . The particle P is in equilibrium when it is 3 m vertically below M, as shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the tension in each string when P is in the equilibrium position.
2. Show that the mass of P is 12.5 kg . The particle P is released from rest at M , and moves in a vertical line.
3. Find the acceleration of P when it is 2.1 m vertically below M .
4. Explain why the maximum speed of P occurs at the equilibrium position.
5. Find the maximum speed of P .

5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h] \end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.

1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
2. Find the mass of the object.
3. Find the elastic potential energy of the string when the object is suspended from it.

\includegraphics{figure_6} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m. The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(AB\) (see diagram).

1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N. [4]

\(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.

1. Find the speed with which \(P\) passes through \(M\). [6]

A particle \(P\) of mass \(0.28 \text{ kg}\) is attached to the mid-point of a light elastic string of natural length \(4 \text{ m}\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and \(4.8 \text{ m}\) apart. \(P\) is released from rest at the mid-point of \(AB\). In the subsequent motion, the acceleration of \(P\) is zero when \(P\) is at a distance \(0.7 \text{ m}\) below \(AB\).

1. Show that the modulus of elasticity of the string is \(20 \text{ N}\). [4]
2. Calculate the maximum speed of \(P\). [3]

\includegraphics{figure_5} A light elastic string has natural length \(2\) m and modulus of elasticity \(\lambda\) N. The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and \(2.4\) m apart. A particle \(P\) of mass \(0.6\) kg is attached to the mid-point of the string and hangs in equilibrium at a point \(0.5\) m below \(AB\) (see diagram).

1. Show that \(\lambda = 26\). [4]

\(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point \(0.9\) m below \(AB\).

1. Calculate the speed of projection of \(P\). [5]

The points \(A\) and \(B\) are at the same horizontal level a distance \(4a\) apart. The ends of a light elastic string, of natural length \(4a\) and modulus of elasticity \(\lambda\), are attached to \(A\) and \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The system is in equilibrium with \(P\) at a distance \(\frac{5}{8}a\) below \(M\), the midpoint of \(AB\).

1. Find \(\lambda\) in terms of \(m\) and \(g\). [3]

The particle \(P\) is pulled down vertically and released from rest at a distance \(\frac{8}{5}a\) below \(M\).

1. Find, in terms of \(a\) and \(g\), the speed of \(P\) as it passes through \(M\) in the subsequent motion. [4]

\includegraphics{figure_4} A small ball of mass \(3m\) is attached to the ends of two light elastic strings \(AP\) and \(BP\), each of natural length \(l\) and modulus of elasticity \(kmg\). The ends \(A\) and \(B\) of the strings are attached to fixed points on the same horizontal level, with \(AB = 2l\). The mid-point of \(AB\) is \(C\). The ball hangs in equilibrium at a distance \(\frac{3}{4}l\) vertically below \(C\) as shown in Figure 4.

1. Show that \(k = 10\) [7]

The ball is now pulled vertically downwards until it is at a distance \(\frac{15}{8}l\) below \(C\). The ball is released from rest.

1. Find the speed of the ball as it reaches \(C\). [6]

Two light elastic strings each have natural length \(0.75\) m and modulus of elasticity \(49\) N. A particle \(P\) of mass \(2\) kg is attached to one end of each string. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 1.5\) m. \includegraphics{figure_2} The particle is held at the mid-point of \(AB\). The particle is released from rest, as shown in Figure 2.

1. Find the speed of \(P\) when it has fallen a distance of \(1\) m. [6]

Given instead that \(P\) hangs in equilibrium vertically below the mid-point of \(AB\), with \(\angle APB = 2\alpha\),

1. show that \(\tan \alpha + 5 \sin \alpha = 5\). [6]

A light elastic string has natural length \(8\) m and modulus of elasticity \(80\) N. The ends of the string are attached to fixed points \(P\) and \(Q\) which are on the same horizontal level and \(12\) m apart. A particle is attached to the mid-point of the string and hangs in equilibrium at a point \(4.5\) m below \(PQ\).

1. Calculate the weight of the particle. [6]
2. Calculate the elastic energy in the string when the particle is in this position. [3]