Small oscillations with elastic strings/springs

2 A particle of mass \(m\) is attached to the mid-point of a light elastic string. The string is stretched between two points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 2 a\). The string has modulus of elasticity \(\lambda\) and natural length \(2 l\), where \(l < a\). The particle is in motion on the surface along a line passing through the mid-point of \(A B\) and perpendicular to \(A B\). When the displacement of the particle from \(A B\) is \(x\), the tension in the string is \(T\). Given that \(x\) is small enough for \(x ^ { 2 }\) to be neglected, show that $$T = \frac { \lambda } { l } ( a - l )$$ The particle is slightly disturbed from its equilibrium position. Show that it will perform approximate simple harmonic motion and find the period of the motion.

2 \includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-2_293_875_525_635} Two light elastic strings, each of natural length \(a\) and modulus of elasticity \(2 m g\), are attached to a particle \(P\) of mass \(m\). The strings join the particle to the points \(A\) and \(B\) which are fixed and at a distance \(4 a\) apart on a smooth horizontal surface. The particle is at rest at the mid-point \(O\) of \(A B\). The particle is now displaced a small distance in a direction perpendicular to \(A B\), on the surface, and released from rest. At time \(t\), the displacement of \(P\) from \(O\) is \(x\) (see diagram). Show that $$\ddot { x } = - \frac { 4 g x } { a } \left( 1 - \frac { 1 } { 2 } \left( 1 + \frac { x ^ { 2 } } { 4 a ^ { 2 } } \right) ^ { - \frac { 1 } { 2 } } \right) .$$ Given that \(\frac { x } { a }\) is so small that \(\left( \frac { x } { a } \right) ^ { 2 }\) and higher powers may be neglected, show that the motion of \(P\) is approximately simple harmonic and state the period of the motion.

7 \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-4_557_1036_278_553} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is pivoted to a fixed point at \(A\) and is free to rotate in a vertical plane. Two fixed vertical wires in this plane are a distance \(6 a\) apart and the point \(A\) is half-way between the two wires. Light smooth rings \(R _ { 1 }\) and \(R _ { 2 }\) slide on the wires and are connected to \(B\) by light elastic strings, each of natural length \(a\) and modulus of elasticity \(\frac { 1 } { 4 } m g\). The strings \(B R _ { 1 }\) and \(B R _ { 2 }\) are always horizontal and the angle between \(A B\) and the upward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a \left( 1 + \cos \theta + \sin ^ { 2 } \theta \right) .$$
2. Given that \(\theta = 0\) is a position of stable equilibrium, find the approximate period of small oscillations about this position.

7 \includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-3_585_801_991_647} A light rod \(A B\) of length \(2 a\) can rotate freely in a vertical plane about a fixed horizontal axis through \(A\). A particle of mass \(m\) is attached to the rod at \(B\). A fixed smooth ring \(R\) lies in the same vertical plane as the rod, where \(A R = a\) and \(A R\) makes an angle \(\frac { 1 } { 4 } \pi\) above the horizontal. A light elastic string, of natural length \(a\) and modulus of elasticity \(m g \sqrt { } 2\), passes through the ring \(R\); one end is fixed to \(A\) and the other end is fixed to \(B\). The rod makes an angle \(\theta\) below the horizontal, where \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\) (see diagram).

1. Use the cosine rule to show that \(R B ^ { 2 } = a ^ { 2 } ( 5 - ( 2 \sqrt { } 2 ) \cos \theta + ( 2 \sqrt { } 2 ) \sin \theta )\).
2. Show that \(\theta = 0\) is a position of stable equilibrium.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - k \sin \theta\), expressing the constant \(k\) in terms of \(a\) and \(g\), and hence write down the approximate period of small oscillations about the equilibrium position \(\theta = 0\).

6 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-3_716_483_890_790} Two small smooth pegs \(P\) and \(Q\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(A\) is the mid-point of \(P Q\). A light rod \(A B\) of length \(4 a\) is freely pivoted at \(A\) and can rotate in the vertical plane containing \(P Q\), with \(B\) below the level of \(P Q\). A particle of mass \(m\) is attached to the rod at \(B\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(\lambda\), passes round the pegs \(P\) and \(Q\) and its two ends are attached to the rod at the point \(X\), where \(A X = a\). The angle between the rod and the downward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram). You are given that the elastic energy stored in the string is \(\lambda a ( 1 + \cos \theta )\).

1. Show that \(\theta = 0\) is a position of equilibrium, and show that the equilibrium is stable if \(\lambda < 4 m g\).
2. Given that \(\lambda = 3 m g\), show that \(\ddot { \theta } = - k \frac { g } { a } \sin \theta\), stating the value of the constant \(k\). Hence find the approximate period of small oscillations of the system about the equilibrium position \(\theta = 0\).

11 Two light strings, each of natural length \(a\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 a\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that $$\ddot { x } = - \frac { 12 g x } { a } \left( 1 - \frac { a } { \sqrt { 9 a ^ { 2 } + x ^ { 2 } } } \right) .$$
2. Given that \(\frac { x } { a }\) is small throughout the motion, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { a }$$ and state the period of the simple harmonic motion that this equation represents.
3. Given that the initial speed of \(P\) is \(\sqrt { \frac { g a } { 200 } }\), show that the amplitude of the simple harmonic motion is \(\frac { 1 } { 40 } a\).

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right) .$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { 1 } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

\includegraphics{figure_2} A bead \(B\) of mass \(m\) is threaded on a smooth circular wire of radius \(r\), which is fixed in a vertical plane. The centre of the circle is \(O\), and the highest point of the circle is \(A\). A light elastic string of natural length \(r\) and modulus of elasticity \(kmg\) has one end attached to the bead and the other end attached to \(A\). The angle between the string and the downward vertical is \(\theta\), and the extension in the string is \(x\), as shown in Figure 2. Given that the string is taut,

1. show that the potential energy of the system is $$2mgr[(k-1)\cos^2 \theta - k\cos \theta] + \text{constant}$$ [6]

Given also that \(k = 3\),

1. find the positions of equilibrium and determine their stability. [9]