6.02i Conservation of energy: mechanical energy principle

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 particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan \theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [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 light elastic string of natural length \(a\) and modulus of elasticity \(2mg\). One end of the string is attached to a fixed point \(A\). The other end of the string is attached to a particle of mass \(2m\).

1. Find, in terms of \(a\), the extension of the string when the particle hangs freely in equilibrium below \(A\). [2]
2. The particle is released from rest at \(A\). Find, in terms of \(a\), the distance of the particle below \(A\) when it first comes to instantaneous rest. [6]

A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is 7 m s\(^{-1}\). The blocks are modelled as particles.

1. Find the value of \(h\). [2] Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
2. find the value of \(R\). [8]

\includegraphics{figure_5} Figure 5 shows two particles \(A\) and \(B\), of mass \(2m\) and \(4m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,

1. give a reason why the magnitudes of the accelerations of the two particles are the same, [1]
2. write down an equation of motion for each particle, [4]
3. find the acceleration of each particle. [5]

Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),

1. find the distance \(XY\) in terms of \(h\). [6]

\includegraphics{figure_3} Particles \(A\) and \(B\), of mass \(2m\) and \(m\) respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle \(A\) is held on the table, while \(B\) rests on a smooth plane inclined at \(30°\) to the horizontal, as shown in Fig. 3. The string is in the same vertical plane as a line of greatest slope of the inclined plane. The coefficient of friction between \(A\) and the table is \(\mu\). The particle \(A\) is released from rest and begins to move. By writing down an equation of motion for each particle,

1. show that, while both particles move with the string taut. Each particle has an acceleration of magnitude \(\frac{1}{5}(1 - 4\mu)g\). [7]

When each particle has moved a distance \(h\), the string breaks. The particle \(A\) comes to rest before reaching the pulley. Given that \(\mu = 0.2\),

1. find, in terms of \(h\), the total distance moved by \(A\). [6]

For the model described above,

1. state two physical factors, apart from air resistance, which could be taken into account to make the model more realistic. [2]

A particle \(P\) of mass \(2\) kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4\) kg which is at rest on the same horizontal plane. Immediately after the collision, \(P\) and \(Q\) are moving in opposite directions and the speed of \(P\) is one-third the speed of \(Q\).

1. Show that the speed of \(P\) immediately after the collision is \(\frac{1}{5}u\) m s\(^{-1}\). [4]

After the collision \(P\) continues to move in the same straight line and is brought to rest by a constant resistive force of magnitude \(10\) N. The distance between the point of collision and the point where \(P\) comes to rest is \(1.6\) m.

1. Calculate the value of \(u\). [5]

\includegraphics{figure_1} Figure 1 A particle \(P\) of mass 6.5 kg is projected up a fixed rough plane with initial speed 6 m s\(^{-1}\) from a point \(X\) on the plane, as shown in Figure 1. The particle moves up the plane along the line of greatest slope through \(X\) and comes to instantaneous rest at the point \(Y\), where \(XY = d\) metres. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{5}{12}\). The coefficient of friction between \(P\) and the plane is \(\frac{1}{3}\).

1. Use the work-energy principle to show that, to 2 significant figures, \(d = 2.7\) [7]

After coming to rest at \(Y\), the particle \(P\) slides back down the plane.

1. Find the speed of \(P\) as it passes through \(X\). [4]

A child is playing with a small model of a fire-engine of mass \(0.5\) kg and a straight, rigid plank. The plank is inclined at an angle \(α\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons. When \(α = 20°\) the fire-engine is projected with an initial speed of \(5\) m s\(^{-1}\) and first comes to rest after travelling 2 m.

1. Find, to 3 significant figures, the value of \(R\). [7]

When \(α = 40°\) the fire-engine is again projected with an initial speed of \(5\) m s\(^{-1}\).

1. Find how far the fire-engine travels before first coming to rest. [3]

A particle of mass 4 kg is moving in a straight horizontal line. There is a constant resistive force of magnitude \(R\) newtons. The speed of the particle is reduced from 25 m s\(^{-1}\) to rest over a distance of 200 m. Use the work-energy principle to calculate the value of \(R\). [4]

\includegraphics{figure_3} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \((-20\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\). Immediately after being struck it has velocity \((15\mathbf{i} + 16\mathbf{j})\) m s\(^{-1}\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.

1. Calculate the magnitude of the impulse exerted by the bat on \(B\). [4]
2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\). [6]
3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\). [4]
4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic. [2]

\includegraphics{figure_3} A particle \(P\) is projected from a point \(A\) with speed \(u\) m s\(^{-1}\) at an angle of elevation \(\theta\), where \(\cos \theta = \frac{4}{5}\). The point \(B\), on horizontal ground, is vertically below \(A\) and \(AB = 45\) m. After projection, \(P\) moves freely under gravity passing through a point \(C\), 30 m above the ground, before striking the ground at the point \(D\), as shown in Figure 3. Given that \(P\) passes through \(C\) with speed 24.5 m s\(^{-1}\),

1. using conservation of energy, or otherwise, show that \(u = 17.5\), [4]
2. find the size of the angle which the velocity of \(P\) makes with the horizontal as \(P\) passes through \(C\), [3]
3. find the distance \(BD\). [7]

A particle of mass \(0.5\) kg is projected vertically upwards from ground level with a speed of \(20 \text{ ms}^{-1}\). It comes to instantaneous rest at a height of \(10\) m above the ground. As the particle moves it is subject to air resistance of constant magnitude \(R\) newtons. Using the work-energy principle, or otherwise, find the value of \(R\). [6]

A cyclist and her cycle have a combined mass of \(75\) kg. The cyclist is cycling up a straight road inclined at \(5°\) to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude \(20\) N. At the instant when the cyclist has a speed of \(12\) m s\(^{-1}\), she is decelerating at \(0.2\) m s\(^{-2}\).

1. Find the rate at which the cyclist is working at this instant. [5]

When the cyclist passes the point \(A\) her speed is \(8\) m s\(^{-1}\). At \(A\) she stops working but does not apply the brakes. She comes to rest at the point \(B\). The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude \(20\) N.

1. Use the work-energy principle to find the distance \(AB\). [5]

A child is playing with a small model of a fire-engine of mass 0.5 kg and a straight, rigid plank. The plank is inclined at an angle \(\alpha\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons. When \(\alpha = 20°\) the fire-engine is projected with an initial speed of 5 m s\(^{-1}\) and first comes to rest after travelling 2 m.

1. Find, to 3 significant figures, the value of \(R\). [7]

When \(\alpha = 40°\) the fire-engine is again projected with an initial speed of 5 m s\(^{-1}\).

1. Find how far the fire-engine travels before first coming to rest. [3]

\includegraphics{figure_1} Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point \(A\) on the top of a hill, she is travelling at 8 m s\(^{-1}\). She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point \(B\) through a vertical distance of 12 m. When she reaches \(B\), her speed is 5 m s\(^{-1}\). The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from \(A\) to \(B\) is 500 m. By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N,

1. find the work done by the cyclist in moving from \(A\) to \(B\). [5]

At \(B\) the road is horizontal. Given that at \(B\) the cyclist is accelerating at 0.5 m s\(^{-2}\),

1. find the power generated by the cyclist at \(B\). [4]

A girl and her bicycle have a combined mass of 64 kg. She cycles up a straight stretch of road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{14}\). She cycles at a constant speed of 5 m s\(^{-1}\). When she is cycling at this speed, the resistance to motion from non-gravitational forces has magnitude 20 N.

1. Find the rate at which the cyclist is working. [4]

She now turns round and comes down the same road. Her initial speed is 5 m s\(^{-1}\), and the resistance to motion is modelled as remaining constant with magnitude 20 N. She free-wheels down the road for a distance of 80 m. Using this model,

1. find the speed of the cyclist when she has travelled a distance of 80 m. [5]

The cyclist again moves down the same road, but this time she pedals down the road. The resistance is now modelled as having magnitude proportional to the speed of the cyclist. Her initial speed is again 5 m s\(^{-1}\) when the resistance to motion has magnitude 20 N.

1. Find the magnitude of the resistance to motion when the speed of the cyclist is 8 m s\(^{-1}\). [1]

The cyclist works at a constant rate of 200 W.

1. Find the magnitude of her acceleration when her speed is 8 m s\(^{-1}\). [4]

A particle \(P\) has mass 4 kg. It is projected from a point \(A\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\frac{2}{5}\). The particle comes to rest instantaneously at the point \(B\) on the plane, where \(AB = 2.5\) m. It then moves back down the plane to \(A\).

1. Find the work done by friction as \(P\) moves from \(A\) to \(B\). [4]

1. Using the work-energy principle, find the speed with which \(P\) is projected from \(A\). [4]

1. Find the speed of \(P\) when it returns to \(A\). [4]

\includegraphics{figure_2} A particle \(P\) of mass 0.5 kg is projected from a point \(A\) up a line of greatest slope \(AB\) of a fixed plane. The plane is inclined at 30° to the horizontal and \(AB = 2\) m with \(B\) above \(A\), as shown in Figure 2. The particle \(P\) passes through \(B\) with speed 5 m s\(^{-1}\). The plane is smooth from \(A\) to \(B\).

1. Find the speed of projection. [4]

The particle \(P\) comes to instantaneous rest at the point \(C\) on the plane, where \(C\) is above \(B\) and \(BC = 1.5\) m. From \(B\) to \(C\) the plane is rough and the coefficient of friction between \(P\) and the plane is \(\mu\). By using the work-energy principle,

1. find the value of \(\mu\). [6]

A ball of mass 0.2 kg is projected vertically upwards from a point \(O\) with speed 20 m s\(^{-1}\). The non-gravitational resistance acting on the ball is modelled as a force of constant magnitude 1.24 N and the ball is modelled as a particle. Find, using the work-energy principle, the speed of the ball when it first reaches the point which is 8 m vertically above \(O\). [6]

\includegraphics{figure_1} The points \(O\) and \(B\) are on horizontal ground. The point \(A\) is \(h\) metres vertically above \(O\). A particle \(P\) is projected from \(A\) with speed 12 m s\(^{-1}\) at an angle \(\alpha°\) to the horizontal. The particle moves freely under gravity and hits the ground at \(B\), as shown in Figure 1. The speed of \(P\) immediately before it hits the ground is 15 m s\(^{-1}\).

1. By considering energy, find the value of \(h\). [4]

Given that 1.5 s after it is projected from \(A\), \(P\) is at a point 4 m above the level of \(A\), find

1. the value of \(\alpha\), [3]
2. the direction of motion of \(P\) immediately before it reaches \(B\). [3]

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{2}{5}\). The particle is held at a point \(A\) on the plane, where \(OA = \frac{5}{4}l\), and released from rest. The particle comes to rest at the point \(B\).

1. Show that \(OB < l\) [4]
2. Find the distance \(OB\). [3]