String becomes slack timing

7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point \(O\). When the particle hangs in equilibrium with the string vertical, the extension of the string is \(e\).

1. Find \(e\).
   (2) The particle is now pulled down a vertical distance \(\frac { 1 } { 3 } a\) below its equilibrium position and released from rest. At time \(t\) after being released, during the time when the string remains taut, the extension of the string is \(e + x\). By forming a differential equation for the motion of \(P\) while the string remains taut,
2. show that during this time \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { a } { 6 g } }\).
   (6)
3. Show that, while the string remains taut, the greatest speed of \(P\) is \(\frac { 1 } { 3 } \sqrt { } ( 6 g a )\).
4. Find \(t\) when the string becomes slack for the first time. \section*{END}

7 A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to one end of a light elastic string of natural length 3.2 m and modulus of elasticity \(4 m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(A\). The particle is released from rest at a point 4.8 m vertically below \(A\). At time \(t \mathrm {~s}\) after \(P\) 's release \(P\) is ( \(4 + x ) \mathrm { m }\) below \(A\).

1. Show that \(4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 49 x\). \(P\) 's motion is simple harmonic.
2. Write down the amplitude of \(P\) 's motion and show that the string becomes slack instantaneously at intervals of approximately 1.8 s . A particle \(Q\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(B\). The particle is released from rest with the string taut and inclined at a small angle with the downward vertical. At time \(t \mathrm {~s}\) after \(Q\) 's release \(B Q\) makes an angle of \(\theta\) radians with the downward vertical.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } \approx - \frac { g } { L } \theta\). The period of the simple harmonic motion to which \(Q\) 's motion approximates is the same as the period of \(P\) 's motion.
4. Given that \(\theta = 0.08\) when \(t = 0\), find the speed of \(Q\) when \(t = 0.25\).

12 A particle P of mass \(m\) is fixed to one end of a light elastic string of natural length \(l\) and modulus of elasticity 12 mg . The other end of the string is attached to a fixed point O . Particle P is held next to O and then released from rest.

1. Show that P next comes instantaneously to rest when the length of the string is \(\frac { 3 } { 2 } l\). The string first becomes taut at time \(t = 0\). At time \(t \geqslant 0\), the length of the string is \(l + x\), where \(x\) is the extension in the string.
2. Show that when the string is taut, \(x\) satisfies the differential equation $$\ddot { \mathrm { x } } + \omega ^ { 2 } \mathrm { x } = \mathrm { g } \text {, where } \omega ^ { 2 } = \frac { 12 \mathrm {~g} } { \mathrm { I } } \text {. }$$
3. By using the substitution \(x = y + \frac { g } { \omega ^ { 2 } }\), solve the differential equation to show that the time when the string first becomes slack satisfies the equation $$\cos \omega \mathrm { t } - \sqrt { \mathrm { k } } \sin \omega \mathrm { t } = 1$$ where \(k\) is an integer to be determined.

1. Throughout this question, use \(\boldsymbol { g } = \mathbf { 1 0 m ~ s } ^ { \mathbf { - 2 } }\)

A light elastic string has natural length 1.25 m and modulus of elasticity 25 N .
A particle \(P\) of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). Particle \(P\) hangs freely in equilibrium with \(P\) vertically below \(A\) The particle is then pulled vertically down to a point \(B\) and released from rest.

1. Show that, while the string is taut, \(P\) moves with simple harmonic motion with period \(\frac { \pi } { \sqrt { 10 } }\) seconds. The maximum kinetic energy of \(P\) during the subsequent motion is 2.5 J .
2. Show that \(A B = 2 \mathrm {~m}\) The particle returns to \(B\) for the first time \(T\) seconds after it was released from rest at \(B\)
3. Find the value of \(T\)

5 A particle of mass \(m\) is attached by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) to a fixed point \(A\), from which it is allowed to fall freely. The particle first comes to rest, instantaneously, at \(B\), where \(A B = 2 l\). Prove that

1. \(\lambda = 4 m g\),
2. while the string is taut, \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g } { l } x\), where \(x\) is the displacement from the equilibrium position at time \(t\),
3. the time taken between the first occasion when the string becomes taut and the next occasion when it becomes slack is $$\left[ \frac { 1 } { 2 } \pi + \sin ^ { - 1 } \left( \frac { 1 } { 3 } \right) \right] \sqrt { \frac { l } { g } }$$

9 A light string, of natural length 0.5 m and modulus of elasticity 4 N , has one end attached to the ceiling of a room. A particle of mass 0.2 kg is attached to the free end of the string and hangs in equilibrium.

1. Find the extension of the string when the particle is in the equilibrium position. The particle is pulled down a further 0.5 m from the equilibrium position and released from rest. At time \(t\) seconds the displacement of the particle from the equilibrium position is \(x \mathrm {~m}\).
2. Show that, while the string is taut, the equation of motion is \(\ddot { x } = - 40 x\).
3. Find the time taken for the string to become slack for the first time.
4. Show that the particle comes to instantaneous rest 0.125 m below the ceiling.

A particle \(P\) of mass \(m\) kg is fixed to one end of a light elastic string of modulus \(mg\) N and natural length \(l\) m. The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(OX = \frac{5l}{4}\) m.

1. Find the coefficient of friction between \(P\) and the table. [2 marks]

\(P\) is now given a small displacement \(x\) m horizontally along \(OX\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.

1. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. [4 marks]

If \(P\) is held at the point where the extension in the string is \(l\) m and then released,

1. show that the string becomes slack after a time \(\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{3}\right)\right)\sqrt{\frac{l}{g}}\) s. [5 marks]
2. Determine the speed of \(P\) when it reaches \(O\). [4 marks]