6.02i Conservation of energy: mechanical energy principle

One end of a light elastic string of natural length \(0.7\) m is attached to a fixed point \(A\) on a smooth horizontal surface. The other end of the string is attached to a particle \(P\) of mass \(0.3\) kg which is held at a point \(B\) on the horizontal surface, where \(AB = 1.2\) m. It is given that \(P\) is released from rest at \(B\) and that when \(AP = 0.9\) m, the particle has speed \(4\) m s\(^{-1}\). Calculate the modulus of elasticity of the string. [3]

A particle \(P\) of mass \(0.3\) kg is attached to one end of a light elastic string of natural length \(0.9\) m and modulus of elasticity \(18\) N. The other end of the string is attached to a fixed point \(O\) which is \(3\) m above the ground.

1. Find the extension of the string when \(P\) is in the equilibrium position. [2]

\(P\) is projected vertically downwards from the equilibrium position with initial speed \(6\) m s\(^{-1}\). At the instant when the tension in the string is \(12\) N the string breaks. \(P\) continues to descend vertically.

1. 1. Calculate the height of \(P\) above the ground at the instant when the string breaks. [2]
   2. Find the speed of \(P\) immediately before it strikes the ground. [4]

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 \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(\theta°\) to the vertical and \(AP = 0.5\) m.

1. Find the angular speed of \(P\) and the value of \(\theta\). [5]
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]

A particle \(P\) of mass \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(θ°\) to the vertical and \(AP = 0.5\) m.

1. Find the angular speed of \(P\) and the value of \(θ\). [5]
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]

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]

\includegraphics{figure_6} A particle \(P\) of mass \(0.2 \text{ kg}\) is attached to one end of a light inextensible string of length \(0.6 \text{ m}\). The other end of the string is attached to a particle \(Q\) of mass \(0.3 \text{ kg}\). The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length \(0.3 \text{ m}\) and modulus of elasticity \(15 \text{ N}\) joins \(Q\) to a fixed point \(A\) which is \(0.4 \text{ m}\) vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended. [4]
2. Find the distance \(HP\) given that the angular speed of \(P\) is \(8 \text{ rad s}^{-1}\). [5]

One end of a light elastic string is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.24 kg. The string has natural length 0.6 m and modulus of elasticity 24 N. The particle is released from rest at \(O\). Find the two possible values of the distance \(OP\) for which the particle has speed 1.5 m s\(^{-1}\). [6]

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]

A particle \(P\) of mass \(0.28 \text{ kg}\) is attached to the mid-point of a light elastic string of natural length \(4 \text{ m}\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and \(4.8 \text{ m}\) apart. \(P\) is released from rest at the mid-point of \(AB\). In the subsequent motion, the acceleration of \(P\) is zero when \(P\) is at a distance \(0.7 \text{ m}\) below \(AB\).

1. Show that the modulus of elasticity of the string is \(20 \text{ N}\). [4]
2. Calculate the maximum speed of \(P\). [3]

\includegraphics{figure_5} A light elastic string has natural length \(2\) m and modulus of elasticity \(\lambda\) N. The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and \(2.4\) m apart. A particle \(P\) of mass \(0.6\) kg is attached to the mid-point of the string and hangs in equilibrium at a point \(0.5\) m below \(AB\) (see diagram).

1. Show that \(\lambda = 26\). [4]

\(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point \(0.9\) m below \(AB\).

1. Calculate the speed of projection of \(P\). [5]

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 \(O\). The other end of the string is attached to a particle \(P\) of mass 0.25 kg. \(P\) hangs in equilibrium below \(O\).

1. Calculate the distance \(OP\). [2]

The particle \(P\) is raised, and is released from rest at \(O\).

1. Calculate the speed of \(P\) when it passes through the equilibrium position. [3]
2. Calculate the greatest value of the distance \(OP\) in the subsequent motion. [3]

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 \(AB = 4\) 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.) [4]
2. Show that the greatest speed of \(P\) occurs when it is \(2.1\) m below \(A\), and calculate this greatest speed. [5]
3. Calculate the greatest magnitude of the acceleration of \(P\). [3]

A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point. The particle moves in a vertical circle.

1. Show that the speed \(v\) at the lowest point of the circle must satisfy \(v^2 \geq 5gl\) for the particle to complete the circle.
2. Given that the particle just completes the circle, find the tensions in the string at the highest and lowest points of the circle.
3. Given that \(v^2 = 6gl\) at the lowest point, find the tension in the string when the particle has risen through an angle \(\theta\) from the lowest point.

[14]

\includegraphics{figure_7} A particle of mass \(0.4\) kg is attached to one end of a light inextensible string of length \(2\) m. The other end of the string is attached to a fixed point \(O\). The particle moves in a vertical circle and passes through the lowest point of the circle with speed \(6\) m s\(^{-1}\).

1. Find the tension in the string when the particle is at the lowest point. [2]
2. Find the speed of the particle when the string makes an angle of \(60°\) with the downward vertical. [4]
3. Hence find the tension in the string at this position. [2]

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]

\includegraphics{figure_7} One end of a light elastic string with modulus of elasticity \(15\) N is attached to a fixed point \(A\) which is \(2\) m vertically above a fixed small smooth ring \(R\). The string has natural length \(2\) m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which moves with constant angular speed \(\omega\) rad s\(^{-1}\) in a horizontal circle which has its centre \(0.4\) m vertically below the ring. \(PR\) makes an acute angle \(\theta\) with the vertical (see diagram).

1. Show that the tension in the string is \(\frac{3}{\cos\theta}\) N and hence find the value of \(m\). [4]
2. Show that the value of \(\omega\) does not depend on \(\theta\). [4]

It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).

1. Find this value of \(\theta\). [4]

A particle \(P\) of mass \(0.2\text{ kg}\) is attached to one end of a light elastic string of natural length \(0.75\text{ m}\) and modulus of elasticity \(21\text{ N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\text{ m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.

1. Find the magnitude of the force exerted on \(P\) by the surface. [2]

\(P\) is now projected horizontally along the surface with speed \(3\text{ m s}^{-1}\).

1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]

A particle \(P\) of mass \(0.3\,\text{kg}\) moves in a straight line on a smooth horizontal surface. \(P\) passes through a fixed point \(O\) of the line with velocity \(8\,\text{m s}^{-1}\). A force of magnitude \(2x\,\text{N}\) acts on \(P\) in the direction \(PO\), where \(x\,\text{m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v\frac{dv}{dx} = kx\) and state the value of the constant \(k\). [2]
2. Find the value of \(x\) at the instant when \(P\) comes to instantaneous rest. [3]

A particle \(P\) of mass \(0.2\,\text{kg}\) is attached to one end of a light elastic string of natural length \(0.75\,\text{m}\) and modulus of elasticity \(21\,\text{N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\,\text{m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.

1. Find the magnitude of the force exerted on \(P\) by the surface. [2]

\(P\) is now projected horizontally along the surface with speed \(3\,\text{m s}^{-1}\).

1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]

A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v\) m s\(^{-1}\) when its displacement is \(x\) m from \(O\). The force acting on \(B\) has magnitude \((2 + 0.3x^2)\) N and is directed horizontally away from \(O\).

1. Show that \(v\frac{dv}{dx} = 1.2x^2 + 8\). [2]
2. Find the velocity of \(B\) when \(x = 1.5\). [3]

An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v\frac{dv}{dx} = 1.2x^2 + 6 - 3x.$$

1. Find the magnitude of this extra force and state the direction in which it acts. [2]

\includegraphics{figure_6} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m. A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg. The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate

1. the speed of \(P\) when it is at the same horizontal level as \(O\), [4]
2. the speed of \(P\) at the instant when the string becomes slack. [3]

A particle \(P\) of mass \(0.4\text{ kg}\) is attached to a fixed point \(O\) by a light elastic string of natural length \(0.5\text{ m}\) and modulus of elasticity \(20\text{ N}\). The particle \(P\) is released from rest at \(O\).

1. Find the greatest speed of \(P\) in the subsequent motion. [4]
2. Find the distance below \(O\) of the point at which \(P\) comes to instantaneous rest. [3]

\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).

1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3]
2. The maximum value of \(v\) is \(4.5\). Find the initial value of \(v\). [5]