6.02i Conservation of energy: mechanical energy principle

A light string has length 1.5 metres. A small sphere is attached to one end of the string. The other end of the string is attached to a fixed point O A thin horizontal bar is positioned 0.9 metres directly below O The bar is perpendicular to the plane in which the sphere moves. The sphere is released from rest with the string taut and at an angle \(\alpha\) to the downward vertical through O The string becomes slack when the angle between the two sections of the string is 60° Ben draws the diagram below to show the initial position of the sphere, the bar and the path of the sphere. \includegraphics{figure_7}

1. State two reasons why Ben's diagram is not a good representation of the situation. [2 marks]
2. Using your answer to part (a), sketch an improved diagram. [1 mark]
3. Find \(\alpha\), giving your answer to the nearest degree. [6 marks]

In this question use \(g = 9.8 \text{ m s}^{-2}\) A lift is used to raise a crate of mass 250 kg The lift exerts an upward force of magnitude \(P\) newtons on the crate. When the crate is at a height of \(x\) metres above its initial position $$P = k(x + 1)(12 - x) + 2450$$ where \(k\) is a constant. The crate is initially at rest, at the point where \(x = 0\)

1. Show that the work done by the upward force as the crate rises to a height of 12 metres is given by $$29400 + 360k$$ [3 marks]
2. The speed of the crate is \(3 \text{ m s}^{-1}\) when it has risen to a height of 12 metres. Find the speed of the crate when it has risen to a height of 15 metres. [5 marks]
3. Find the height of the crate when its speed becomes zero. [2 marks]
4. Air resistance has been ignored. Explain why this is reasonable in this context. [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\) A light elastic string has natural length 3 metres and modulus of elasticity 18 newtons. One end of the elastic string is attached to a particle of mass 0.25 kg The other end of the elastic string is attached to a fixed point \(O\) The particle is released from rest at a point \(A\), which is 4.5 metres vertically below \(O\)

1. Calculate the elastic potential energy of the string when the particle is at \(A\) [2 marks]
2. The point \(B\) is 3 metres vertically below \(O\) Calculate the gravitational potential energy gained by the particle as it moves from \(A\) to \(B\) [2 marks]
3. Find the speed of the particle at \(B\) [3 marks]
4. The point \(C\) is 3.6 metres vertically below \(O\) Explain, showing any calculations that you make, why the speed of the particle is increasing the first time that the particle is at \(C\) [3 marks]

\includegraphics{figure_2} A smooth wire is shaped into a circle of centre \(O\) and radius 0.8 m. The wire is fixed in a vertical plane. A small bead \(P\) of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of \(4.2 \, \text{m s}^{-1}\). When \(OP\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \, \text{m s}^{-1}\) (see diagram).

1. Show that \(v^2 = 33.32 - 15.68\cos\theta\). [4]
2. Prove that the bead is never at rest. [1]
3. Find the maximum value of \(v\). [2]

One end of a light inextensible string of length \(0.8\) m is attached to a particle \(P\) of mass \(m\) kg. The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3\) m s\(^{-1}\) so that it moves in a vertical circular path with centre \(O\) (see diagram). \includegraphics{figure_1} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac{1}{3}\pi\) radians with the downward vertical through \(O\).

1. Show that at this instant the speed of \(P\) is \(4.5\) m s\(^{-1}\). [3]
2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant. [3]
3. Find the magnitude of the tangential acceleration of \(P\) at this instant. [2]

One end of a light elastic string of natural length \(2.1\) m and modulus of elasticity \(4.8\) N is attached to a particle, \(P\), of mass \(1.75\) kg. The other end of the string is attached to a fixed point, \(O\), which is on a rough inclined plane. The angle between the plane and the horizontal is \(\theta\) where \(\sin\theta = \frac{3}{5}\). The coefficient of friction between \(P\) and the plane is \(0.732\). Particle \(P\) is placed on the plane at \(O\) and then projected down a line of greatest slope of the plane with an initial speed of \(2.4\) m s\(^{-1}\). Determine the distance that \(P\) has travelled from \(O\) at the instant when it first comes to rest. You can assume that during its motion \(P\) does not reach the bottom of the inclined plane. [8]

\includegraphics{figure_6} The rim of a smooth hemispherical bowl is a circle of centre O and radius \(a\). The bowl is fixed with its rim horizontal and uppermost. A particle P of mass \(m\) is released from rest at a point A on the rim as shown in Fig. 6. When P reaches the lowest point of the bowl it collides directly with a stationary particle Q of mass \(\frac{1}{2}m\). After the collision Q just reaches the rim of the bowl. Find the coefficient of restitution between P and Q. [7]

\includegraphics{figure_9} A particle P of mass \(m\) is joined to a fixed point O by a light inextensible string of length \(l\). P is released from rest with the string taut and making an acute angle \(\alpha\) with the downward vertical, as shown in Fig. 9. At a time \(t\) after P is released the string makes an angle \(\theta\) with the downward vertical and the tension in the string is \(T\). Angles \(\alpha\) and \(\theta\) are measured in radians.

1. Show that $$\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^2 = \frac{2g}{l}\cos\theta + k_1,$$ where \(k_1\) is a constant to be determined in terms of \(g\), \(l\) and \(\alpha\). [4]
2. Show that $$T = 3mg\cos\theta + k_2,$$ where \(k_2\) is a constant to be determined in terms of \(m\), \(g\) and \(\alpha\). [3]

It is given that \(\alpha\) is small enough for \(\alpha^2\) to be negligible.

1. Find, in terms of \(m\) and \(g\), the approximate tension in the string. [2]
2. Show that the motion of P is approximately simple harmonic. [3]

\includegraphics{figure_10} A small toy car runs along a track in a vertical plane. The track consists of three sections: a curved section AB, a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius \(r\) m. The section BC is tangential to the curved section at B and tangential to the circular section at C, as shown in the diagram. The car, of mass \(m\) kg, is placed on the track at A, at a height \(h\) m above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C. It can be assumed that the track does not obstruct the car moving on to the circular section at C. The track is modelled as being smooth, and the car is modelled as a particle P.

1. Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is $$mg\left(3\cos\theta - 2 + \frac{2h}{r}\right)\text{N},$$ where \(\theta\) is the angle between OC and OP. [7]
2. Hence determine, in terms of \(r\), the least possible value of \(h\) so that P can complete a vertical circle. [2]
3. Apart from not modelling the car as a particle, state one refinement that would make the model more realistic. [1]

One end of a rope is attached to a block A of mass 2 kg. The other end of the rope is attached to a second block B of mass 4 kg. Block A is held at rest on a fixed rough ramp inclined at \(30°\) to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P, at a distance \(d\) m above the ground, as shown in the diagram. \includegraphics{figure_7} Block A is more than \(d\) m from P. The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is \(\frac{1}{2\sqrt{3}}\). The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.

1. Determine, in terms of \(g\) and \(d\), the work done against friction as A moves \(d\) m up the ramp. [3]
2. Given that the speed of B immediately before it hits the ground is \(1.75 \text{ m s}^{-1}\), use the work–energy principle to determine the value of \(d\). [5]
3. Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic. [1]

\includegraphics{figure_10} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O. The point A lies on the rim of the bowl such that AO makes an angle of \(30°\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A, with speed \(u\), where \(u > \sqrt{\frac{gr}{2}}\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A. The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.

1. Determine, in terms of \(m\), \(g\), \(u\), \(r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B. [7]

The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4mg\).

1. Determine, in terms of \(r\), the vertical distance of B above the floor. [4]

It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.

1. Show that \(u^2 > 2gr\). [5]

Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u_1 \mathbf{i} + u_2 \mathbf{j}\) towards B, which is at rest. The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf{i}\). The coefficient of restitution between A and B is \(e\).

1. 1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf{i}\). [1]
   2. Determine this impulse in terms of \(\lambda\), \(m\), \(e\) and \(u_1\). [6]

The loss in kinetic energy due to the collision between A and B is \(\frac{1}{8}mu_1^2\).

1. Determine the range of possible values of \(\lambda\). [6]

A car of mass 850 kg is travelling along a straight horizontal road. The power developed by the car is constant and is equal to 18 kW. There is a constant resistance to motion of magnitude 600 N.

1. Find the greatest steady speed at which the car can travel. [2]

Later in the journey, while travelling at a speed of \(15 \text{ m s}^{-1}\), the car comes to the bottom of a straight hill which is inclined at an angle of \(\sin^{-1}\left(\frac{1}{40}\right)\) to the horizontal. The power developed by the car remains constant at 18 kW. The magnitude of the resistance force is no longer constant but changes such that the total work done against the resistance force in ascending the hill is 103 000 J. The car takes 10 seconds to ascend the hill and at the top of the hill the car is travelling at \(18 \text{ m s}^{-1}\).

1. Determine the distance the car travels from the bottom to the top of the hill. [5]

A particle P of mass \(3m\) kg is attached to one end of a light elastic string of modulus of elasticity \(4mg\) N and natural length 0.4 m. The other end of the string is attached to a fixed point O. The particle P rests in equilibrium at a point A with the string vertical.

1. Find the distance OA. [2]

At time \(t = 0\) seconds, P is given a speed of \(2.5 \text{ m s}^{-1}\) vertically downwards from A.

1. Show that P initially performs simple harmonic motion with amplitude \(a\) m, where \(a\) is to be determined correct to 3 significant figures. [5]
2. Determine the smallest distance between P and O in the subsequent motion. [3]

Two small uniform discs A and B, of equal radius, have masses 3 kg and 5 kg respectively. The discs are sliding on a smooth horizontal surface and collide obliquely. The contact between the discs is smooth and A is stationary after the collision. Immediately before the collision B is moving with speed \(2 \text{ m s}^{-1}\) in a direction making an angle of \(60°\) with the line of centres, XY (see diagram below). \includegraphics{figure_12}

1. Explain how you can tell that A must have been moving along XY before the collision. [1]

The coefficient of restitution between A and B is 0.8.

1. • Determine the speed of A immediately before the collision. • Determine the speed of B immediately after the collision. [7]
2. Determine the angle turned through by the direction of B in the collision. [3]

Disc B subsequently collides with a smooth wall, which is parallel to XY. The kinetic energy of B after the collision with the wall is 95% of the kinetic energy of B before the collision with the wall.

1. Determine the coefficient of restitution between B and the wall. [4]

A particle P of mass \(0.5\) kg is attached to a fixed point O by a light elastic string of natural length \(3\) m and modulus of elasticity \(75\) N. P is released from rest at O and is allowed to fall freely. Determine the length of the string when P is at its lowest point in the subsequent motion. [5]

\includegraphics{figure_7} A particle P of mass \(m\) is attached to one end of a light elastic string of natural length \(6a\) and modulus of elasticity \(3mg\). The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of \(30°\) to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A, as shown in Fig. 7. P is now pulled a further distance \(2a\) down the line of greatest slope through A and released from rest. At time \(t\) later, the displacement of P from A is \(x\), where the positive direction of \(x\) is down the plane.

1. Show that, until the string slackens, \(x\) satisfies the differential equation $$\frac{d^2x}{dt^2} + \frac{gx}{2a} = 0.$$ [6]
2. Determine, in terms of \(a\) and \(g\), the time at which the string slackens. [5]
3. Find, in terms of \(a\) and \(g\), the speed of P when the string slackens. [2]

\includegraphics{figure_10} Fig. 10 shows a small bead P of mass \(m\) which is threaded on a smooth thin wire. The wire is in the form of a circle of radius \(a\) and centre O. The wire is fixed in a vertical plane. The bead is initially at the lowest point A of the wire and is projected along the wire with a velocity which is just sufficient to carry it to the highest point on the wire. The angle between OP and the downward vertical is denoted by \(\theta\).

1. Determine the value of \(\theta\) when the magnitude of the reaction of the wire on the bead is \(\frac{7}{5}mg\). [7]
2. Show that the angular velocity of P when OP makes an angle \(\theta\) with the downward vertical is given by \(k\sqrt{\frac{g}{a}\cos\left(\frac{\theta}{2}\right)}\), stating the value of the constant \(k\). [4]
3. Hence determine, in terms of \(g\) and \(a\), the angular acceleration of P when \(\theta\) takes the value found in part (a). [3]

A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\). For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4. \includegraphics{figure_4}

1. By considering the energy of P, show that \(v^2 = u^2 + 2ga(1 - \cos\theta)\). [2]
2. Show that the magnitude of the normal contact force between the sphere and particle P is $$mg(3\cos\theta - 2) - \frac{mv^2}{a}.$$ [2]

The particle loses contact with the sphere when \(\cos\theta = \frac{3}{4}\).

1. Find an expression for \(u\) in terms of \(a\) and \(g\). [2]

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]

A particle \(P\), of mass \(m\) kg, is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). Initially, \(P\) is held at rest with the string just taut and making an angle of 60° with the downward vertical. It is then given a velocity \(u\text{ ms}^{-1}\) perpendicular to the string in a downward direction.

1. 1. When the string makes an angle \(\theta\) with the downward vertical, the velocity of the particle is \(v\) and the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(l\), \(v^2\) and \(\theta\).
   2. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > 4lg\). [10]
2. Given that now \(u^2 = 3lg\), find the position of the string when circular motion ceases. Briefly describe the motion of \(P\) after circular motion has ceased. [3]
3. The string is replaced by a light rigid rod. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > klg\), where \(k\) is to be determined. [2]

The diagram below shows a woman standing at the end of a diving platform. She is about to dive into the water below. The woman has mass 60 kg and she may be modelled as a particle positioned at the end of the platform which is 10 m above the water. \includegraphics{figure_2} When the woman dives, she projects herself from the platform with a speed of \(7.8\text{ ms}^{-1}\).

1. Find the kinetic energy of the woman when she leaves the platform. [2]
2. Initially, the situation is modelled ignoring air resistance. By using conservation of energy, show that the model predicts that the woman enters the water with an approximate speed of \(16\text{ ms}^{-1}\). [6]
3. Suppose that this model is refined to include air resistance so that the speed with which the woman enters the water is now predicted to be \(13\text{ ms}^{-1}\). Determine the amount of energy lost to air resistance according to the refined model. [3]

A particle \(P\) of mass 0.5 kg is in equilibrium under the action of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\). $$\mathbf{F}_1 = (9\mathbf{i} + 6\mathbf{j} - 12\mathbf{k})\text{N} \quad \text{and} \quad \mathbf{F}_2 = (6\mathbf{i} - 7\mathbf{j} + 3\mathbf{k})\text{N}.$$

1. Find the force \(\mathbf{F}_3\). [2]
2. Forces \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are removed so that \(P\) moves in a straight line \(AB\) under the action of the single force \(\mathbf{F}_1\). The points \(A\) and \(B\) have position vectors \((2\mathbf{i} - 9\mathbf{j} + 7\mathbf{k})\) m and \((8\mathbf{i} - 5\mathbf{j} - \mathbf{k})\) m respectively. The particle \(P\) is initially at rest at \(A\).
   1. Verify that \(\mathbf{F}_1\) acts parallel to the vector \(\overrightarrow{AB}\).
   2. Find the work done by the force \(\mathbf{F}_1\) as the particle moves from \(A\) to \(B\).
   3. By using the work-energy principle, find the speed of \(P\) as it reaches \(B\). [7]

One end of a light elastic string, of natural length 2.5 m and modulus of elasticity \(30g\) N, is fixed to a point O. A particle \(P\), of mass 2 kg, is attached to the other end of the string. Initially, \(P\) is held at rest at the point O. It is then released and allowed to fall under gravity.

1. Show that, while the string is taut, $$v^2 = g(5 + 2x - 6x^2),$$ where \(v\text{ ms}^{-1}\) denotes the velocity of the particle when the extension in the string is \(x\) m. [6]
2. Calculate the maximum extension of the string. [3]
3. 1. Find the extension of the string when \(P\) attains its maximum speed.
   2. Hence determine the maximum speed of \(P\). [5]

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]