Vertical SHM with two strings

5. \begin{figure}[h] \end{figure} The fixed points, \(A\) and \(B\), are a distance \(10 a\) apart, with \(B\) vertically above \(A\). One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(2 m g\), is attached to a particle \(P\) of mass \(m\) and the other end is attached to \(A\). One end of another light elastic string, of natural length \(4 a\) and modulus of elasticity \(6 m g\), is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(C\), as shown in Figure 6.

1. Show that each string has an extension of \(2 a\).
   (5) The particle \(P\) is now pulled down vertically, so that it is a distance \(a\) below \(C\) and then released from rest.
2. Show that in the subsequent motion, \(P\) performs simple harmonic motion.
3. Find, in terms of \(a\) and \(g\), the speed of \(P\) when it is a distance \(\frac { 7 } { 2 } a\) above \(A\).

1. The fixed point \(A\) is vertically above the fixed point \(B\), with \(A B = 3 l\)

A light elastic string has natural length \(l\) and modulus of elasticity \(4 m g\) One end of the string is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m\) A second light elastic string also has natural length \(l\) and modulus of elasticity \(4 m g\) One end of this string is attached to \(P\) and the other end is attached to \(B\). Initially \(P\) rests in equilibrium at the point \(E\), where \(A E B\) is a vertical straight line.

1. Show that \(A E = \frac { 13 } { 8 } l\) The particle \(P\) is now held at the point that is a distance \(2 l\) vertically below \(A\) and released from rest. At time \(t\), the vertical displacement of \(P\) from \(E\) is \(x\), where \(x\) is measured vertically downwards.
2. Show that \(\ddot { x } = - \frac { 8 g } { l } x\)
3. Find, in terms of \(g\) and \(l\), the speed of \(P\) when it is \(\frac { 1 } { 8 } l\) below \(E\).
4. Find the length of time, in each complete oscillation, for which \(P\) is more than \(1.5 l\) from \(A\), giving your answer in terms of \(g\) and \(l\)

3 A block of mass 200 kg is connected to a horizontal ceiling by four identical light elastic ropes, each having natural length 7 m and stiffness \(180 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). It is also connected to the floor by a single light elastic rope having stiffness \(80 \mathrm { Nm } ^ { - 1 }\). Throughout this question you may assume that all five ropes are stretched and vertical, and you may neglect air resistance. \begin{figure}[h] \end{figure} Fig. 3 shows the block resting in equilibrium, with each of the top ropes having length 10 m and the bottom rope having length 8 m .

1. Find the tension in one of the top ropes.
2. Find the natural length of the bottom rope. The block now moves vertically up and down. At time \(t\) seconds, the block is \(x\) metres below its equilibrium position.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\). The motion is started by pulling the block down 2.2 m below its equilibrium position and releasing it from rest. The block then executes simple harmonic motion with amplitude 2.2 m .
4. Find the maximum magnitude of the acceleration of the block.
5. Find the speed of the block when it has travelled 3.8 m from its starting point.
6. Find the distance travelled by the block in the first 5 s .

3 A fixed point A is 12 m vertically above a fixed point B. A light elastic string, with natural length 3 m and modulus of elasticity 1323 N , has one end attached to A and the other end attached to a particle P of mass 15 kg . Another light elastic string, with natural length 4.5 m and modulus of elasticity 1323 N , has one end attached to B and the other end attached to P .

1. Verify that, in the equilibrium position, \(\mathrm { AP } = 5 \mathrm {~m}\). The particle P now moves vertically, with both strings AP and BP remaining taut throughout the motion. The displacement of P above the equilibrium position is denoted by \(x \mathrm {~m}\) (see Fig. 3). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-4_405_360_751_849} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Show that the tension in the string AP is \(441 ( 2 - x ) \mathrm { N }\) and find the tension in the string BP .
3. Show that the motion of P is simple harmonic, and state the period. The minimum length of AP during the motion is 3.5 m .
4. Find the maximum length of AP .
5. Find the speed of P when \(\mathrm { AP } = 4.1 \mathrm {~m}\).
6. Find the time taken for AP to increase from 3.5 m to 4.5 m .

6. Two points \(A\) and \(B\) are in a vertical line, with \(A\) above \(B\) and \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) hangs at rest in equilibrium.

1. Show that \(A P = \frac { 7 a } { 4 }\) The particle \(P\) is now pulled down vertically from its equilibrium position towards \(B\) and at time \(t = 0\) it is released from rest. At time \(t\), the particle \(P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. The particle \(P\) is subject to air resistance of magnitude \(m k v\), where \(k\) is a positive constant.
2. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
3. Find the range of values of \(k\) which would result in the motion of \(P\) being a damped oscillation.

7 \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-4_622_767_269_689} Particles \(P\) and \(Q\), with masses \(3 m\) and \(2 m\) respectively, are connected by a light inextensible string passing over a smooth light pulley. The particle \(P\) is connected to the floor by a light spring \(S _ { 1 }\) with natural length \(a\) and modulus of elasticity mg . The particle \(Q\) is connected to the floor by a light spring \(S _ { 2 }\) with natural length \(a\) and modulus of elasticity \(2 m g\). The sections of the string not in contact with the pulley, and the two springs, are vertical. Air resistance may be neglected. The particles \(P\) and \(Q\) move vertically and the string remains taut; when the length of \(S _ { 1 }\) is \(x\), the length of \(S _ { 2 }\) is ( \(3 a - x\) ) (see diagram).

1. Find the total potential energy of the system (taking the floor as the reference level for gravitational potential energy). Hence show that \(x = \frac { 4 } { 3 } a\) is a position of stable equilibrium.
2. By differentiating the energy equation, and substituting \(x = \frac { 4 } { 3 } a + y\), show that the motion is simple harmonic, and find the period.

1. The points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 4.5 \mathrm {~m}\).

A light elastic string has natural length 1.5 m and modulus of elasticity 15 N . One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). A particle, \(P\), of mass 0.2 kg , is attached to the stretched string so that \(A P B\) is a straight line and \(A P = 1.5 \mathrm {~m}\). The particle rests in equilibrium on the surface. The particle is now moved directly towards \(A\) and is held on the surface so \(A P B\) is a straight line with \(A P = 1 \mathrm {~m}\). The particle is released from rest.

1. Prove that \(P\) moves with simple harmonic motion.
2. Find
   1. the maximum speed of \(P\) during the motion,
   2. the maximum acceleration of \(P\) during the motion.
3. Find the total time, in each complete oscillation of \(P\), for which the speed of \(P\) is greater than \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

\includegraphics{figure_3} A small object \(P\) is attached to one end of each of two vertical light elastic strings. One string is of natural length \(0.4\) m and has modulus of elasticity \(10\) N; the other string is of natural length \(0.5\) m and has modulus of elasticity \(12\) N. The upper ends of both strings are attached to a fixed horizontal beam and \(P\) hangs in equilibrium \(0.6\) m below the beam (see diagram).

1. Show that the weight of \(P\) is \(7.4\) N and find the total elastic potential energy stored in the two strings when \(P\) is hanging in equilibrium. [6]

\(P\) is then held at a point \(0.7\) m below the beam with the strings vertical. \(P\) is released from rest.

1. Show that, throughout the subsequent motion, \(P\) performs simple harmonic motion, and find the period. [7]