Vertical elastic string: released from rest, string starts taut

1 One end of a light elastic string of natural length 1.6 m and modulus of elasticity 25 N is attached to a fixed point \(A\). A particle \(P\) of mass 0.15 kg is attached to the other end of the string. \(P\) is held at rest at a point 2 m vertically below \(A\) and is then released.

1. For the motion from the instant of release until the string becomes slack, find the loss of elastic potential energy and the gain in gravitational potential energy.
2. Hence find the speed of \(P\) at the instant the string becomes slack.

1 A light elastic string has modulus of elasticity 5 N and natural length 1.5 m . One end of the string is attached to a fixed point \(O\) and a particle \(P\) of mass 0.1 kg is attached to the other end of the string. \(P\) is released from rest at the point 2.4 m vertically below \(O\). Calculate the speed of \(P\) at the instant the string first becomes slack.

One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(5mg\), is attached to a fixed point \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The spring hangs with \(P\) vertically below \(A\). The particle \(P\) is released from rest in the position where the extension of the spring is \(\frac{3}{5}a\).

1. Show that the initial acceleration of \(P\) is \(\frac{3}{5}g\) upwards. [3]
2. Find the speed of \(P\) when the spring first returns to its natural length. [4]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3mg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2a\). The particle \(P\) is then released from rest.

1. Find the speed of \(P\) when it is at a distance \(\frac{3}{4}a\) below \(O\). [3]
2. Find the initial acceleration of \(P\) when it is released from rest. [2]

A light elastic string has natural length \(a\) and modulus of elasticity \(12mg\). One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(e > \frac{1}{4}a\). In the subsequent motion the particle has speed \(\sqrt{2ga}\) when it has ascended a distance \(\frac{1}{4}a\). Find \(e\) in terms of \(a\). [6]

A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(5mg\). The other end of the spring is attached to a fixed point \(O\). The spring hangs vertically with \(P\) below \(O\). The particle \(P\) is pulled down vertically and released from rest when the length of the spring is \(\frac{7}{5}a\). Find the distance of \(P\) below \(O\) when \(P\) first comes to instantaneous rest. [4]

A light elastic string of natural length \(1.5\) m and modulus of elasticity \(490\) N has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(30\) kg. Initially, \(P\) is held at rest vertically below \(A\) such that the distance \(AP\) is \(0.6\) m. It is then allowed to fall vertically.

1. Calculate the distance \(AP\) when \(P\) is instantaneously at rest for the first time, giving your answer correct to 2 decimal places. [8]
2. Estimate the distance \(AP\) when \(P\) is instantaneously at rest for the second time and clearly state one assumption that you have made in making your estimate. [2]

One end of a light elastic string, of natural length \(0.2\) m and modulus of elasticity \(5g\) N, is attached to a fixed point \(O\). The other end is attached to a particle of mass \(4\) kg. The particle hangs in equilibrium vertically below \(O\).

1. Show that the extension of the string is \(0.16\) m. [2]
2. The particle is pulled down vertically and held at rest so that the extension of the string is \(0.28\) m. The particle is then released. Determine the speed of the particle as it passes through the equilibrium position. [8]