Complete motion cycle with slack phase

6. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } \mathrm { mg }\). A particle \(P\) of mass \(m\) 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 at the point \(O\), vertically below \(A\).

1. Find the distance \(O A\). The particle \(P\) is now pulled vertically down to a point \(B\), where \(A B = 3 a\), and released from rest.
2. Show that, throughout the subsequent motion, \(P\) performs only simple harmonic motion, justifying your answer. The point \(C\) is vertically below \(A\), where \(A C = 2 a\).
   Find, in terms of \(a\) and \(g\),
3. the speed of \(P\) at the instant that it passes through \(C\),
4. the time taken for \(P\) to move directly from \(B\) to \(C\). \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-17_2255_50_314_34}
   

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1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\). The other end of the string is attached to a fixed point on a ceiling. The particle \(P\) hangs in equilibrium at a distance \(D\) below the ceiling.

The particle \(P\) is now pulled vertically downwards until it is a distance \(3 l\) below the ceiling and released from rest. Given that \(P\) comes to instantaneous rest just before it reaches the ceiling,

1. show that \(D = \frac { 5 l } { 3 }\)
2. Show that, while the elastic string is stretched, \(P\) moves with simple harmonic motion, with period \(2 \pi \sqrt { \frac { 2 l } { 3 g } }\)
3. Find, in terms of \(g\) and \(l\), the exact time from the instant when \(P\) is released to the instant when the elastic string first goes slack.

7. A particle \(P\) of mass 2 kg is attached to one end of a light elastic string, of natural length 1 m and modulus of elasticity 98 N . The other end of the string is attached to a fixed point \(A\). When \(P\) hangs freely below \(A\) in equilibrium, \(P\) is at the point \(E , 1.2 \mathrm {~m}\) below \(A\). The particle is now pulled down to a point \(B\) which is 0.4 m vertically below \(E\) and released from rest.

1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion about \(E\) with period \(\frac { 2 \pi } { 7 } \mathrm {~s}\).
2. Find the greatest magnitude of the acceleration of \(P\) while the string is taut.
3. Find the speed of \(P\) when the string first becomes slack.
4. Find, to 3 significant figures, the time taken, from release, for \(P\) to return to \(B\) for the first time.

7. A particle \(B\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N . The other end of the string is attached to a fixed point \(A\). The particle is hanging in equilibrium at the point \(E\), vertically below \(A\).

1. Show that \(A E = 0.9 \mathrm {~m}\). The particle is held at \(A\) and released from rest. The particle first comes to instantaneous rest at the point \(C\).
2. Find the distance \(A C\).
3. Show that while the string is taut, \(B\) is moving with simple harmonic motion with centre \(E\).
4. Calculate the maximum speed of \(B\).

7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string. The string has natural length \(l\) metres and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point \(A\). The particle hangs freely in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = 1.4 \mathrm {~m}\).

1. Show that \(l = 1.2\) The point \(C\) is vertically below \(A\) and \(A C = 1.8 \mathrm {~m}\). The particle is pulled down to \(C\) and released from rest.
2. Show that, while the string is taut, \(P\) moves with simple harmonic motion.
3. Calculate the speed of \(P\) at the instant when the string first becomes slack. The particle first comes to instantaneous rest at the point \(D\).
4. Find the time taken by \(P\) to return directly from \(D\) to \(C\).
   

One end of a light elastic string is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and hangs freely under gravity. In the equilibrium position, the extension of the string is \(d\). Show that the period of small vertical oscillations about the equilibrium position is \(2 \pi \sqrt { } \left( \frac { d } { g } \right)\). The particle is now pulled down and released from rest at a distance \(2 d\) below the equilibrium position. Given that the particle does not reach \(O\) in the subsequent motion, show that the time taken until the particle first comes to instantaneous rest is \(\left( \sqrt { } 3 + \frac { 2 } { 3 } \pi \right) \sqrt { } \left( \frac { d } { g } \right)\).

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4 m g\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac { 1 } { 8 } l\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi \sqrt { } \left( \frac { l } { g } \right)\). At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac { 7 } { 16 } l\) under gravity, it strikes a fixed smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac { 1 } { 3 }\). Show that the speed of \(P\) immediately after the impact is \(\frac { 1 } { 4 } \sqrt { } ( 5 g l )\).

A light elastic string, of natural length \(l\) and modulus of elasticity \(4 m g\), is attached at one end to a fixed point and has a particle \(P\) of mass \(m\) attached to the other end. When \(P\) is hanging in equilibrium under gravity it is given a velocity \(\sqrt { } ( g l )\) vertically downwards. At time \(t\) the downward displacement of \(P\) from its equilibrium position is \(x\). Show that, while the string is taut, $$\ddot { x } = - \frac { 4 g } { l } x .$$ Find the speed of \(P\) when the length of the string is \(l\). Show that the time taken for \(P\) to move from the lowest point to the highest point of its motion is $$\left( \frac { \pi } { 3 } + \frac { \sqrt { } 3 } { 2 } \right) \sqrt { } \left( \frac { l } { g } \right)$$

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(8 m g\) and natural length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is pulled vertically downwards a distance \(\frac { 1 } { 4 } a\) from its equilibrium position and released from rest. Show that the string first becomes slack after a time \(\frac { 2 \pi } { 3 } \sqrt { } \left( \frac { a } { 8 g } \right)\). Find, in terms of \(a\), the total distance travelled by \(P\) from its release until it subsequently comes to instantaneous rest for the first time.

6. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\frac { m g } { 2 }\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(O E = ( l + e ) \mathrm { m }\)

1. Find the numerical value of the ratio \(e : l\).
   \(P\) is now pulled down a further distance \(\frac { 3 l } { 2 } \mathrm {~m}\) from \(E\) and is released from rest.
   In the subsequent motion, the string remains taut. At time \(t \mathrm {~s}\) after being released, \(P\) is at a distance \(x \mathrm {~m}\) below \(E\).
2. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic.
3. Write down the period of the motion.
4. Find the speed with which \(P\) first passes through \(E\) again.
5. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where $$A E = \frac { 3 l } { 4 } \mathrm {~m} , \text { is } \frac { 2 \pi } { 3 } \sqrt { \frac { 2 l } { g } } \mathrm {~s} .$$

7. One end of a light elastic string, of natural length \(3 l \mathrm {~m}\), is attached to a fixed point \(O\). A particle of mass \(m \mathrm {~kg}\) is attached to the other end of the string. When the particle hangs freely in equilibrium, the string is extended by a length of \(l \mathrm {~m}\). The particle is then pulled down through a further distance \(2 l \mathrm {~m}\) and released from rest.

1. Prove that as long as the string is taut, the particle performs simple harmonic motion about its equilibrium position.
2. Show that the time between the release of the particle and the instant when the string becomes slack is \(\frac { 2 } { 3 } \pi \sqrt { \frac { l } { g } } \mathrm {~s}\).
3. Find the greatest height reached by the particle above its point of release.
4. Show that the time \(T\) s taken to reach this greatest height from the moment of release is given by \(T = \left( \frac { 2 \pi } { 3 } + \sqrt { 3 } \right) \sqrt { \frac { l } { g } }\).
   (4 marks)

7 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 39.2 mN . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Show that, while the string is in tension, the particle performs simple harmonic motion about a point 1 m below \(O\).
2. Show that when \(P\) is at its lowest point the extension of the string is 0.8 m .
3. Find the time after its release that \(P\) first reaches its lowest point.
4. Find the velocity of \(P 0.8 \mathrm {~s}\) after it is released from \(O\). OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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5. A toy car of mass 0.5 kg is attached to one end \(A\) of a light elastic string \(A B\), of natural length 1.5 m and modulus of elasticity 27 N . Initially the car is at rest on a smooth horizontal floor and the string lies in a straight line with \(A B = 1.5 \mathrm {~m}\). The end \(B\) is moved in a straight horizontal line directly away from the car, with constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds after \(B\) starts to move, the extension of the string is \(x\) metres and the car has moved a distance \(y\) metres. The effect of air resistance on the car can be ignored. By modelling the car as a particle, show that, while the string remains taut,

1. 1. \(x + y = u t\)
   2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 36 x = 0\)
2. Hence show that the string becomes slack when \(t = \frac { \pi } { 6 }\)
3. Find, in terms of \(u\), the speed of the car when \(t = \frac { \pi } { 12 }\)
4. Find, in terms of \(u\), the distance the car has travelled when it first reaches end \(B\) of the string.