Elastic string with variable force

4 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 12 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.24 \mathrm {~kg} . P\) is projected vertically upwards with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 0.8 m vertically below \(O\).

1. Calculate the speed of the particle when it is moving upwards with zero acceleration.
2. Show that the particle moves 0.6 m while it is moving upwards with constant acceleration.

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 19.2 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) is released from rest at the point 2.7 m vertically above \(A\). Calculate

1. the initial acceleration of \(P\),
2. the speed of \(P\) when it reaches \(A\).

7 A particle \(P\) of mass 0.5 kg is attached to a fixed point \(O\) by a light elastic string of natural length 1 m and modulus of elasticity 16 N . The particle \(P\) is projected vertically upwards from \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 x ^ { 2 } \mathrm {~N}\) acts on \(P\) when \(P\) has displacement \(x \mathrm {~m}\) above \(O\). After projection the upwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, before the string becomes taut, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.2 x ^ { 2 }\).
2. Find the velocity of \(P\) at the instant the string becomes taut.
3. Find an expression for the acceleration of \(P\) while it is moving upwards after the string becomes taut.
4. Verify that \(P\) comes to instantaneous rest before the extension of the string is 0.5 m .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.8 m vertically below \(O\). When the extension of the string is \(x \mathrm {~m}\), the downwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and a force of magnitude \(25 x ^ { 2 } \mathrm {~N}\) opposes the motion of \(P\).

1. Show that, when \(P\) is moving downwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 40 x - 50 x ^ { 2 }\).
2. For the instant when \(P\) has its greatest downwards speed, find the kinetic energy of \(P\) and the elastic potential energy stored in the string.

1 One end of a light elastic rope of natural length 2.5 m and modulus of elasticity 80 N is attached to a fixed point \(A\). A stone \(S\) of mass 8 kg is attached to the other end of the rope. \(S\) is held at a point 6 m vertically below \(A\) and then released. Find the initial acceleration of \(S\).

7 A light elastic string has natural length 3 m and modulus of elasticity 45 N . A particle \(P\) of weight 6 N is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\) and \(A B = 4 \mathrm {~m}\). The particle \(P\) is released from rest at the point 1.5 m vertically below \(A\).

1. Calculate the distance \(P\) moves after its release before first coming to instantaneous rest at a point vertically above \(B\). (You may assume that at this point the part of the string joining \(P\) to \(B\) is slack.)
2. Show that the greatest speed of \(P\) occurs when it is 2.1 m below \(A\), and calculate this greatest speed.
3. Calculate the greatest magnitude of the acceleration of \(P\).

5 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.3 \mathrm {~kg} . P\) is projected vertically upwards with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 1.2 m vertically below \(O\).

1. Calculate the speed of the particle at the position where it is moving with zero acceleration. [5
2. Show that the particle moves 1.2 m while moving upwards with constant deceleration.

3 A particle \(P\) of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point \(O\). After projection the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(8 x \mathrm {~N}\) directed away from \(O\) acts on \(P\) and a force of magnitude ( \(2 \mathrm { e } ^ { - x } + 4\) ) N opposes the motion of \(P\). One end of a light elastic string of natural length 0.5 m is attached to \(O\) and the other end of the string is attached to \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 20 x - 10 - 5 \mathrm { e } ^ { - x }\) before the elastic string becomes taut.
2. Given that the initial velocity of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(v\) when the string first becomes taut.
   When the string is taut, the acceleration of \(P\) is proportional to \(\mathrm { e } ^ { - x }\).
3. Find the modulus of elasticity of the string.