Elastic string on smooth inclined plane

3 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point \(A\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg which rests in equilibrium on the plane.

1. Calculate the extension of the string. \(P\) is projected down the plane from the equilibrium position with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The extension of the string is \(e \mathrm {~m}\) when the speed of the particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the first time.
2. Find \(e\).

5 A light elastic string has natural length 2 m and modulus of elasticity 1.5 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.075 \mathrm {~kg} . P\) is released from rest at \(O\). Find

1. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
2. the acceleration with which \(P\) starts to move up the plane immediately after it has reached its lowest point.

6 A light elastic string has natural length 4 m and modulus of elasticity 2 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.1 \mathrm {~kg} . P\) is held at rest at \(O\) and then released. The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(x \mathrm {~m}\).

1. Show that \(v ^ { 2 } = 45 - 5 ( x - 1 ) ^ { 2 }\). Hence find
2. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
3. the maximum speed of \(P\).

7 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a particle \(P\) of mass 0.8 kg . The other end of the string is attached to a fixed point \(O\) at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The particle rests in equilibrium on the plane.

1. Calculate the extension of the string. \(P\) is projected from its equilibrium position up the plane along a line of greatest slope. In the subsequent motion \(P\) just reaches \(O\), and later just reaches the foot of the plane. Calculate
2. the speed of projection of \(P\),
3. the length of the line of greatest slope of the plane.

2. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(l\) and modulus of elasticity \(\lambda\) One end of the spring is attached to a point \(A\) on a smooth plane.
The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) A particle \(P\) of mass \(m\) is attached to the other end of the spring. Initially \(P\) is held at the point \(B\) on the plane, where \(A B\) is a line of greatest slope of the plane. The point \(B\) is lower than \(A\) and \(A B = 2 l\), as shown in Figure 1 .
The particle is released from rest at \(B\) and first comes to instantaneous rest at the point \(C\) on \(A B\), where \(A C = 0.7 l\)

1. Use the principle of conservation of mechanical energy to show that $$\lambda = \frac { 100 } { 91 } m g$$
2. Find the acceleration of \(P\) when it is released from rest at \(B\).

4. \begin{figure}[h] \end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\), is fixed to a point \(A\) on a smooth plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the elastic string. Initially, \(B\) is held at rest at the point \(C\) on the plane with the elastic string lying along a line of greatest slope of the plane. The point \(C\) is below \(A\) and \(A C = l\), as shown in Figure 2 . The ball is released and comes to instantaneous rest at a point \(D\) on the plane.
The points \(A , C\) and \(D\) all lie along a line of greatest slope of the plane and \(A D = \frac { 5 l } { 4 }\) The ball is modelled as a particle and air resistance is modelled as being negligible.
Using the model,

1. show that \(\lambda = 4 \mathrm { mg }\)
2. find, in terms of \(g\) and \(l\), the greatest speed of \(B\) as it moves from \(C\) to \(D\)

1. A particle \(P\) of mass \(m\) lies on a smooth plane inclined at an angle \(30 ^ { \circ }\) to the horizontal. The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m g\). The other end of the string is attached to a fixed point \(O\) on the plane. The particle \(P\) is in equilibrium at the point \(A\) on the plane and the extension of the string is \(\frac { 1 } { 4 } a\). The particle \(P\) is now projected from \(A\) down a line of greatest slope of the plane with speed \(V\). It comes to instantaneous rest after moving a distance \(\frac { 1 } { 2 } a\).

By using the principle of conservation of energy,

1. find \(V\) in terms of \(a\) and \(g\),
2. find, in terms of \(a\) and \(g\), the speed of \(P\) when the string first becomes slack.

5. \begin{figure}[h] \end{figure} One end \(A\) of a light elastic string, of natural length \(a\) and modulus of elasticity \(6 m g\), is fixed at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the string. Initially \(B\) is held at rest with the string lying along a line of greatest slope of the plane, with \(B\) below \(A\) and \(A B = a\). The ball is released and comes to instantaneous rest at a point \(C\) on the plane, as shown in Figure 2. Find

1. the length \(A C\),
2. the greatest speed attained by \(B\) as it moves from its initial position to \(C\).

1. \begin{figure}[h] \end{figure} A particle of mass 0.8 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N . The other end of the spring is attached to a fixed point \(O\) on a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held on the plane at a point which is 1.6 m down a line of greatest slope of the plane from \(O\), as shown in Figure 1. The particle is then released from rest. Find the initial acceleration of the particle.
(Total 6 marks)

6 A light elastic string has one end attached to a point \(A\) fixed on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle of mass 6 kg . The elastic string has natural length 4 metres and modulus of elasticity 300 newtons. The particle is pulled down the plane in the direction of the line of greatest slope through \(A\). The particle is released from rest when it is 5.5 metres from \(A\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_314_713_1900_660}

1. Calculate the elastic potential energy of the string when the particle is 5.5 metres from the point \(A\).
2. Show that the speed of the particle when the string becomes slack is \(3.66 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
3. Show that the particle will not reach point \(A\) in the subsequent motion.

5 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 0.75 mgN . The other end of the string is attached to a fixed point \(O\) of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is released from rest at \(O\) and moves down the plane.

1. Show that the maximum speed of \(P\) is reached when the extension of the string is 0.8 m .
2. Find the maximum speed of \(P\).
3. Find the maximum displacement of \(P\) from \(O\).

5 One end of a light elastic string, of natural length 0.78 m and modulus of elasticity 0.8 mgN , is attached to a fixed point \(O\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. \(P\) is released from rest at \(O\) and moves down the plane without reaching the bottom. Find

1. the maximum speed of \(P\) in the subsequent motion,
2. the distance of \(P\) from \(O\) when it is at its lowest point.

A particle \(P\) is attached to one end of a light elastic string of natural length \(1.2 \text{ m}\) and modulus of elasticity \(12 \text{ N}\). The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. \(P\) rests in equilibrium on the plane, \(1.6 \text{ m}\) from \(O\).

1. Calculate the mass of \(P\). [2]

A particle \(Q\), with mass equal to the mass of \(P\), is projected up the plane along a line of greatest slope. When \(Q\) strikes \(P\) the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving \(0.2 \text{ m}\).

1. Show that the initial kinetic energy of the combined particle is \(1 \text{ J}\). [4]

The combined particle subsequently moves down the plane.

1. Calculate the greatest speed of the combined particle in the subsequent motion. [5]

A particle \(P\) is attached to one end of a light elastic string of natural length \(1.2\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. \(P\) rests in equilibrium on the plane, \(1.6\) m from \(O\).

1. Calculate the mass of \(P\). [2]

A particle \(Q\), with mass equal to the mass of \(P\), is projected up the plane along a line of greatest slope. When \(Q\) strikes \(P\) the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving \(0.2\) m.

1. Show that the initial kinetic energy of the combined particle is \(1\) J. [4]

The combined particle subsequently moves down the plane.

1. Calculate the greatest speed of the combined particle in the subsequent motion. [5]

A particle \(P\) of mass \(0.7 \text{ kg}\) is attached by a light elastic string to a fixed point \(O\) on a smooth plane inclined at an angle of \(30°\) to the horizontal. The natural length of the string is \(0.5 \text{ m}\) and the modulus of elasticity is \(20 \text{ N}\). The particle \(P\) is projected up the line of greatest slope through \(O\) from a point \(A\) below the level of \(O\). The initial kinetic energy of \(P\) is \(1.8 \text{ J}\) and the initial elastic potential energy in the string is also \(1.8 \text{ J}\).

1. Find the distance \(OA\). [2]
2. Find the greatest speed of \(P\) in the motion. [6]

One end of a light elastic string of natural length \(0.8\) m and modulus of elasticity \(36\) N is attached to a fixed point \(O\) on a smooth plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{3}{5}\). A particle \(P\) of mass \(2\) kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of \(O\). The particle is projected with speed \(\sqrt{2}\) m s\(^{-1}\) directly down the plane from the position where \(OP\) is equal to the natural length of the string. Find the maximum extension of the string during the subsequent motion. [5]

\includegraphics{figure_4} A light spring of natural length \(a\) and modulus of elasticity \(kmg\) is attached to a fixed point \(O\) on a smooth plane inclined to the horizontal at an angle \(\theta\), where \(\sin\theta = \frac{1}{4}\). A particle of mass \(m\) is attached to the lower end of the spring and is held at the point \(A\) on the plane, where \(OA = 2a\) and \(OA\) is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed \(V\) when it passes through the point \(B\) on the plane, where \(OB = \frac{3}{2}a\). The speed of the particle is \(\frac{1}{3}V\) when it passes through the point \(C\) on the plane, where \(OC = \frac{3}{4}a\). Find the value of \(k\). [7]

A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3mg\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. [3]

A particle of mass \(m\) kg is attached to the end \(B\) of a light elastic string \(AB\). The string has natural length \(l\) m and modulus of elasticity \(\lambda\) N. \includegraphics{figure_3} The end \(A\) is attached to a fixed point on a smooth plane inclined at an angle \(\alpha\) to the horizontal, as shown, and the particle rests in equilibrium with the length \(AB = \frac{5l}{4}\) m.

1. Show that \(\lambda = 4 mg \sin \alpha\). [3 marks]

The particle is now moved and held at rest at \(A\) with the string slack. It is then gently released so that it moves down the plane along a line of greatest slope.

1. Find the greatest distance from \(A\) that the particle reaches down the plane. [6 marks]

Use \(g\) as 9.81 m s\(^{-2}\) in this question. A light elastic string has one end attached to a fixed point A on a smooth plane inclined at 25° to the horizontal. The other end of the string is attached to a wooden block of mass 2.5 kg, which rests on the plane. The elastic string has natural length 3 metres and modulus of elasticity 125 newtons. The block is pulled down the line of greatest slope of the plane to a point 4.5 metres from A and then released.

1. Find the elastic potential energy of the string at the point when the block is released. [1 mark]
2. Calculate the speed of the block when the string becomes slack. [4 marks]
3. Determine whether the block reaches the point A in the subsequent motion, commenting on any assumptions that you make. [3 marks]