Simple harmonic motion with elastic string

6. A particle of mass \(m\) is attached to one end of a light elastic string, of natural length \(6 a\) and modulus of elasticity 9 mg . The other end of the string is attached to a fixed point \(A\) on a ceiling. The particle hangs in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = ( 6 + p ) a\).

1. Show that \(p = \frac { 2 } { 3 }\) The particle is now released from rest at a point \(C\) vertically below \(B\), where \(A C < \frac { 22 } { 3 } a\).
2. Show that the particle moves with simple harmonic motion.
3. Find the period of this motion.
4. Explain briefly the significance of the condition \(A C < \frac { 22 } { 3 } a\). The point \(D\) is vertically below \(A\) and \(A D = 8 a\). The particle is now released from rest at \(D\). The particle first comes to instantaneous rest at the point \(E\).
5. Find, in terms of \(a\), the distance \(A E\).

5. \begin{figure}[h] \end{figure} Two fixed points \(A\) and \(B\) are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\). A second light elastic string, of natural length 1.2 m and modulus of elasticity 15 N , has one end attached to \(P\) and the other end attached to \(B\). Initially \(P\) rests in equilibrium at the point \(O\), as shown in Figure 3.

1. Show that \(A O = 3 \mathrm {~m}\). The particle is now pulled towards \(A\) and released from rest at the point \(C\), where \(A C B\) is a straight line and \(O C = 1 \mathrm {~m}\).
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion.
3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. The particle first comes to instantaneous rest at the point \(D\).
4. Find the distance \(D B\).

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The particle \(P\) is at rest at the point \(B\) on the table, where \(A B = l\).

At time \(t = 0 , P\) is projected along the table with speed \(U\) in the direction \(A B\).
At time \(t\)

• the elastic string has not gone slack
• \(B P = x\)
• the speed of \(P\) is \(v\)
  1. Show that

$$v ^ { 2 } = U ^ { 2 } - \frac { 2 g x ^ { 2 } } { l }$$

By differentiating this equation with respect to \(x\), prove that, before the elastic string goes slack, \(P\) moves with simple harmonic motion with period \(\pi \sqrt { \frac { 2 l } { g } }\) Given that \(U = \sqrt { \frac { g l } { 2 } }\)

find, in terms of \(l\) and \(g\), the exact total time, from the instant it is projected from \(B\), that it takes \(P\) to travel a total distance of \(\frac { 3 } { 4 } l\) along the table.

5. \begin{figure}[h] \end{figure} Two fixed points \(A\) and \(B\) are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\). A second light elastic string, of natural length 1.2 m and modulus of elasticity 15 N , has one end attached to \(P\) and the other end attached to \(B\). Initially \(P\) rests in equilibrium at the point \(O\), as shown in Figure 3.

1. Show that \(A O = 3 \mathrm {~m}\). The particle is now pulled towards \(A\) and released from rest at the point \(C\), where \(A C B\) is a straight line and \(O C = 1 \mathrm {~m}\).
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion.
3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. The particle first comes to instantaneous rest at the point \(D\).
4. Find the distance \(D B\).

5. A light elastic string of natural length \(l\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached tot he other end of the string and hangs in equilibrium at the point \(O\), where \(A O = \frac { 5 } { 4 } l\).

1. Find the modulus of the elasticity of the string. The particle \(P\) is then pulled down and released from rest. At time \(t\) the length of the string is \(\frac { 5 l } { 4 } + x\).
2. Prove that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g x } { l }$$ When \(P\) is released, \(A P = \frac { 7 } { 4 } l\). The point \(B\) is a distance \(l\) vertically below \(A\).
3. Find the speed of \(P\) at \(B\).
4. Describe briefly the motion of \(P\) after it has passed through \(B\) for the first time until it next passes through \(O\).

5 \includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-3_378_625_1272_758} A light elastic band, of total natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\), is stretched over two small smooth pins fixed at the same horizontal level and at a distance \(a\) apart. A particle of mass \(m\) is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle \(\beta\) with the vertical (see diagram). Express the tension in the band in terms of \(m , g\) and \(\beta\), and hence show that \(\beta = \frac { 1 } { 4 } \pi\). The particle is given a velocity of magnitude \(\sqrt { } ( a g )\) vertically downwards. At time \(t\) the displacement of the particle from its equilibrium position is \(x\). Show that, neglecting air resistance, $$\ddot { x } = - \frac { 2 g } { a } x .$$ Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time.

3 A light elastic rope has natural length 15 m . One end of the rope is attached to a fixed point O and the other end is attached to a small rock of mass 12 kg . When the rock is hanging in equilibrium vertically below O , the length of the rope is 15.8 m .

1. Show that the modulus of elasticity of the rope is 2205 N . The rock is pulled down to the point 20 m vertically below O , and is released from rest in this position. It moves upwards, and comes to rest instantaneously, with the rope slack, at the point A .
2. Find the acceleration of the rock immediately after it is released.
3. Use an energy method to find the distance OA. At time \(t\) seconds after release, the rope is still taut and the displacement of the rock below the equilibrium position is \(x\) metres.
4. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 12.25 x\).
5. Write down an expression for \(x\) in terms of \(t\), and hence find the time between releasing the rock and the rope becoming slack.

In this question take \(g = 10\). A particle P of mass 0.15 kg is attached to one end of a light elastic string of modulus of elasticity 13.5 N and natural length 0.45 m. The other end of the string is attached to a fixed point O. The particle P rests in equilibrium at a point A with the string vertical.

1. Show that the distance OA is 0.5 m. [3]

At time \(t = 0\), P is projected vertically downwards from A with a speed of 1.25 m s\(^{-1}\). Throughout the subsequent motion, P experiences a variable resistance \(R\) newtons which is of magnitude 0.6 times its speed (in m s\(^{-1}\)).

1. Given that the downward displacement of P from A at time \(t\) seconds is \(x\) metres, show that, while the string remains taut, \(x\) satisfies the differential equation $$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 200x = 0.$$ [3]
2. Verify that \(x = \frac{5}{56}e^{-2t}\sin(14t)\). [6]
3. Determine whether the string becomes slack during the motion. [5]