Vertical elastic string: projected from equilibrium or other point

5 A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(A\), and \(P\) hangs in equilibrium.

1. Calculate the extension of the string. \(P\) is projected vertically downwards from the equilibrium position with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance \(A P\) when the speed of \(P\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is below the equilibrium position.
3. Calculate the speed of \(P\) when it is 0.5 m above the equilibrium position.

5 A particle \(P\) of mass 0.7 kg is attached to a fixed point \(O\) by a light elastic string of natural length 0.6 m and modulus of elasticity 15 N . The particle \(P\) is projected vertically downwards from the point \(A , 0.8 \mathrm {~m}\) vertically below \(O\). The initial speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the distance below \(A\) of the point at which \(P\) comes to instantaneous rest.
2. Find the greatest speed of \(P\) in the motion. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-10_478_652_260_751} The diagram shows a uniform lamina \(A B C D E F G H\). The lamina consists of a quarter-circle \(O A B\) of radius \(r \mathrm {~m}\), a rectangle \(D E F G\) and two isosceles right-angled triangles \(C O D\) and \(G O H\). The rectangle has \(D G = E F = r \mathrm {~m}\) and \(D E = F G = x \mathrm {~m}\).
3. Given that the centre of mass of the lamina is at \(O\), express \(x\) in terms of \(r\).
4. Given instead that the rectangle \(D E F G\) is a square with edges of length \(r \mathrm {~m}\), state with a reason whether the centre of mass of the lamina lies within the square or the quarter-circle. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-12_384_693_258_726} A rough horizontal rod \(A B\) of length 0.45 m rotates with constant angular velocity \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis through \(A\). A small ring \(R\) of mass 0.2 kg can slide on the rod. A particle \(P\) of mass 0.1 kg is attached to the mid-point of a light inextensible string of length 0.6 m . One end of the string is attached to \(R\) and the other end of the string is attached to \(B\), with angle \(R P B = 60 ^ { \circ }\) (see diagram). \(R\) and \(P\) move in horizontal circles as the system rotates. \(R\) is in limiting equilibrium.
5. Show that the tension in the portion \(P R\) of the string is 1.66 N , correct to 3 significant figures.
6. Find the coefficient of friction between the ring and the rod.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).

1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
   1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
   2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.

1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\). One end of the string is attached to a fixed point \(A\) on a horizontal ceiling. The other end is attached to a particle \(P\) of mass \(m\).

The particle \(P\) hangs in equilibrium at the point \(E\), where \(A E = 3 a\).
The particle \(P\) is then projected vertically downwards from \(E\) with speed \(\frac { 3 } { 2 } \sqrt { a g }\) Air resistance is assumed to be negligible.
Find the elastic energy stored in the string, when \(P\) first comes to instantaneous rest. Give your answer in the form kmga, where \(k\) is a constant to be found.

7 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).

1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
   1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
   2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.

One end of a light elastic string of natural length \(0.6 \text{ m}\) and modulus of elasticity \(24 \text{ N}\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.4 \text{ kg}\) which hangs in equilibrium vertically below \(O\).

1. Calculate the extension of the string. [2]

\(P\) is projected vertically downwards from the equilibrium position with speed \(5 \text{ m s}^{-1}\).

1. Calculate the distance \(P\) travels before it is first at instantaneous rest. [4]

When \(P\) is first at instantaneous rest a stationary particle of mass \(0.4 \text{ kg}\) becomes attached to \(P\).

1. Find the greatest speed of the combined particle in the subsequent motion. [4]

One end of a light elastic string of natural length \(1.6\) m and modulus of elasticity \(28\) N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.35\) kg which hangs in equilibrium vertically below \(O\). The particle \(P\) is projected vertically upwards from the equilibrium position with speed \(1.8\) m s\(^{-1}\). Calculate the speed of \(P\) at the instant the string first becomes slack. [5]

A particle \(P\) of mass \(m\text{kg}\) is attached to one end of a light elastic string of natural length \(2\text{m}\) and modulus of elasticity \(2mg\text{N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d\text{m}\) below its equilibrium position and released from rest.

1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]

A particle \(P\) of mass \(m \text{ kg}\) is attached to one end of a light elastic string of natural length \(2 \text{ m}\) and modulus of elasticity \(2mg \text{ N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d \text{ m}\) below its equilibrium position and released from rest.

1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]

3 A light elastic string has natural length 0.8 m and modulus of elasticity 16 N . One end of the string is attached to a fixed point \(O\), and a particle \(P\) of mass 0.4 kg is attached to the other end of the string. The particle \(P\) hangs in equilibrium vertically below \(O\).

1. Show that the extension of the string is 0.2 m . \(P\) is projected vertically downwards from the equilibrium position. \(P\) first comes to instantaneous rest at the point where \(O P = 1.4 \mathrm {~m}\).
2. Calculate the speed at which \(P\) is projected.
3. Find the speed of \(P\) at the first instant when the string subsequently becomes slack.