6.02i Conservation of energy: mechanical energy principle

A uniform rod \(AB\), of length \(2a\) and mass \(2m\), can rotate freely in a vertical plane about a smooth horizontal axis through \(A\). A small rough ring of mass \(m\) is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a\dot{\theta}^2 = \frac{18}{11}g \sin \theta,$$ where \(\theta\) is the angle turned through by the rod. [3] Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are \(\frac{20}{11}mg \sin \theta\) and \(\frac{2}{11}mg \cos \theta\) respectively. [6] The coefficient of friction between the ring and the rod is \(\mu\). Find, in terms of \(\mu\), the value of \(\theta\) when the ring starts to slide. [2]

Answer only one of the following two alternatives. EITHER A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), is attached at one end to a fixed point and has a particle \(P\) of mass \(m\) attached to the other end. When \(P\) is hanging in equilibrium under gravity it is given a velocity \(\sqrt{(gl)}\) vertically downwards. At time \(t\) the downward displacement of \(P\) from its equilibrium position is \(x\). Show that, while the string is taut, $$\ddot{x} = -\frac{4g}{l}x.$$ [4] Find the speed of \(P\) when the length of the string is \(l\). [4] Show that the time taken for \(P\) to move from the lowest point to the highest point of its motion is $$\left(\frac{\pi}{3} + \frac{\sqrt{3}}{2}\right)\sqrt{\left(\frac{l}{g}\right)}.$$ [6] OR \includegraphics{figure_11} The scatter diagram shows a sample of size 5 of bivariate data, together with the regression line of \(y\) on \(x\). State what is minimised in obtaining this regression line, illustrating your answer on a copy of this diagram. [2] State, giving a reason, whether, for the data shown, the regression line of \(y\) on \(x\) is the same as the regression line of \(x\) on \(y\). [1] A car is travelling along a stretch of road with speed \(v\) km h\(^{-1}\) when the brakes are applied. The car comes to rest after travelling a further distance of \(z\) m. The values of \(z\) (and \(\sqrt{z}\)) for 8 different values of \(v\) are given in the table, correct to 2 decimal places.

\(v\)	25	30	35	40	45	50	55	60
\(z\)	2.83	4.63	4.84	5.29	9.73	10.30	14.82	15.21
\(\sqrt{z}\)	1.68	2.15	2.20	2.30	3.12	3.21	3.85	3.90

[\(\sum v = 340\), \(\sum v^2 = 15500\), \(\sum \sqrt{z} = 22.41\), \(\sum z = 67.65\), \(\sum v\sqrt{z} = 1022.15\).]

1. Calculate the product moment correlation coefficient between \(v\) and \(\sqrt{z}\). What does this indicate about the scatter diagram of the points \((v, \sqrt{z})\)? [4]
2. Given that the product moment correlation coefficient between \(v\) and \(z\) is 0.965, correct to 3 decimal places, state why the regression line of \(\sqrt{z}\) on \(v\) is more suitable than the regression line of \(z\) on \(v\), and find the equation of the regression line of \(\sqrt{z}\) on \(v\). [5]
3. Comment, in the context of the question, on the value of the constant term in the equation of the regression line of \(\sqrt{z}\) on \(v\). [2]

A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac{1}{2}u\) the angle \(\theta\) between \(OP\) and the downward vertical satisfies the equation $$8ga(1 - \cos\theta) = 3u^2.$$ [2] Find, in terms of \(m\), \(u\), \(a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position. [4]

A uniform circular disc has diameter \(AB\), mass \(2m\) and radius \(a\). A particle of mass \(m\) is attached to the disc at \(B\). The disc is able to rotate about a smooth fixed horizontal axis through \(A\). The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is \(\frac{5}{2}ma^2\). [4] The disc is held with \(AB\) horizontal and released. Find the angular speed of the system when \(B\) is directly below \(A\). [5] The disc is slightly displaced from the position of equilibrium in which \(B\) is below \(A\). At time \(t\) the angle between \(AB\) and the vertical is \(\theta\). Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position. [5]

Answer only one of the following two alternatives. **EITHER** A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N, and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N. The mid-point of \(AB\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(MO = 0.2\) m. [4] The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac{\mathrm{d}^2y}{\mathrm{d}t^2} = -500y,$$ where \(y\) metres is the displacement from \(O\) in the direction \(OB\) at time \(t\) seconds, and state the period of the motion. [5] For the instant when the particle is 0.3 m from \(M\) for the first time, find

1. the speed of the particle, [2]
2. the time taken, after release, to reach this position. [3]

**OR** The continuous random variable \(T\) has a negative exponential distribution with probability density function given by $$\mathrm{f}(t) = \begin{cases} \lambda\mathrm{e}^{-\lambda t} & t \geqslant 0, \\ 0 & \text{otherwise.} \end{cases}$$ Show that for \(t \geqslant 0\) the distribution function is given by F\((t) = 1 - \mathrm{e}^{-\lambda t}\). [2] The table below shows some values of F\((t)\) for the case when the mean is 20. Find the missing value. [2] It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below. Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. [3] Perform a \(\chi^2\)-test of goodness of fit, at the 5% level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions. [7]

\includegraphics{figure_3} A smooth cylinder of radius \(a\) is fixed with its axis horizontal. The point \(O\) is the centre of a circular cross-section of the cylinder. The line \(AOB\) is a diameter of this circular cross-section and the radius \(OA\) makes an angle \(\alpha\) with the upward vertical (see diagram). It is given that \(\cos \alpha = \frac{3}{5}\). A particle \(P\) of mass \(m\) moves on the inner surface of the cylinder in the plane of the cross-section. The particle passes through \(A\) with speed \(u\) along the surface in the downwards direction. The magnitude of the reaction between \(P\) and the inner surface of the sphere is \(R_A\) when \(P\) is at \(A\), and is \(R_B\) when \(P\) is at \(B\). It is given that \(R_B = 10R_A\). Show that \(u^2 = ag\). [6] The particle loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\). [4]

The points \(A\) and \(B\) are on a smooth horizontal table at a distance \(8a\) apart. A particle \(P\) of mass \(m\) lies on the table on the line \(AB\), between \(A\) and \(B\). The particle is attached to \(A\) by a light elastic string of natural length \(3a\) and modulus of elasticity \(6mg\), and to \(B\) by a light elastic string of natural length \(2a\) and modulus of elasticity \(mg\). In equilibrium, \(P\) is at the point \(O\) on \(AB\).

1. Show that \(AO = 3.6a\). [4]

The particle is released from rest at the point \(C\) on \(AB\), between \(A\) and \(B\), where \(AC = 3.4a\).

1. Show that \(P\) moves in simple harmonic motion and state the period. [6]
2. Find the greatest speed of \(P\). [2]

\(A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(AB = 3a\) m. One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(mg\) N, is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m\) kg. One end of a second light elastic string, of natural length \(ka\) m and modulus of elasticity \(2mg\) N, is attached to \(B\). The other end of this string is attached to \(P\). Given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(AB\), find the value of \(k\). [3] The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion. [5] In the case where \(P\) is released from rest at a distance \(0.2a\) m from \(M\), the speed of \(P\) is \(0.7\) m s\(^{-1}\) when \(P\) is \(0.05a\) m from \(M\). Find the value of \(a\). [3]

\includegraphics{figure_3} A uniform disc, of radius \(a\) and mass \(2M\), is attached to a thin uniform rod \(AB\) of length \(6a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).

1. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc. [4]

The object is free to rotate about the axis \(l\). The object is held with \(AB\) horizontal and is released from rest. When \(AB\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac{3}{5}\), the angular speed of the object is \(\sqrt{\left(\frac{2g}{5a}\right)}\).

1. Find the possible values of \(x\). [5]

The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(AB = 2.6\) m. One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(0.6\) N, is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg. One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.62\) N, is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).

1. Show that \(AE = 1.4\) m. [4]

The particle \(P\) is now displaced slightly from \(E\), along the line \(AB\).

1. Show that, in the subsequent motion, \(P\) performs simple harmonic motion. [5]
2. Given that the period of the motion is \(\frac{4}{\pi}\) s, find the value of \(\lambda\). [3]

Answer only one of the following two alternatives. **EITHER** One end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 40 N, is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses 2 kg and \(M\) kg attached to its free end. The \(M\) kg particle becomes detached from the spring, and as a result the 2 kg particle begins to move upwards. \begin{enumerate}[label=(\roman*)] \item Show that the 2 kg particle performs simple harmonic motion about its equilibrium position with period \(\frac{2\pi}{5}\) s. State the distance below \(O\) of the centre of the oscillations. [7] \item The speed of the 2 kg particle is 0.4 m s\(^{-1}\) when its displacement from the centre of oscillation is 0.06 m. Find the amplitude of the motion. [3] \item Deduce the value of \(M\). [4] \end{enumerate] **OR** In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. \(\Sigma x = 10.56\) \(\quad\) \(\Sigma x^2 = 14.1775\) \(\quad\) \(\Sigma y = 12.39\) \(\quad\) \(\Sigma y^2 = 15.894\) A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\). \begin{enumerate}[label=(\roman*)] \item Test, at the 10% significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal. [9] \item A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as \(\Sigma x = 10.24\) \(\quad\) and \(\quad\) \(\Sigma(x - \bar{x})^2 = 0.294\), where \(\bar{x}\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p\) kg. A test of this claim is carried out at the 10% significance level, using only this second sample from lake \(A\). This test supports the scientist's claim. Find the greatest possible value of \(p\). [5] \end{enumerate]

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 \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt{(2ag)}\) so that it begins to move along a circular path. The string becomes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos \theta = \frac{2}{3}\). [5]
2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. [4]

Answer only one of the following two alternatives. EITHER The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac{2}{3}\) kg is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to \(P\) and the other end attached to \(B\).

1. Show that when \(P\) is in equilibrium \(AP = 0.75\) m. [3]

The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.

1. Show that \(P\) performs simple harmonic motion and state the period of the motion. [6]
2. Find the speed of \(P\) when it passes through the equilibrium position. [2]
3. Find the speed of \(P\) when its acceleration is equal to half of its maximum value. [3]

OR The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.

Number of repairs in a week	0	1	2	3	4	5	\(\geqslant 6\)
Number of weeks	6	15	9	6	3	1	0

1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data. [3]

Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.

Number of repairs in a week	0	1	2	3	4	5	\(\geqslant 6\)
Expected frequency	8.076	12.921	10.337	5.513	2.205	\(a\)	\(b\)

1. Show that \(a = 0.706\) and find the value of the constant \(b\). [3]
2. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level. [8]

A child of mass \(35\text{ kg}\) is swinging on a rope. The child is modelled as a particle \(P\) and the rope is modelled as a light inextensible string of length \(4\text{ m}\). Initially \(P\) is held at an angle of \(45°\) to the vertical (see diagram). \includegraphics{figure_5}

1. Given that there is no resistance force, find the speed of \(P\) when it has travelled half way along the circular arc from its initial position to its lowest point. [4]
2. It is given instead that there is a resistance force. The work done against the resistance force as \(P\) travels from its initial position to its lowest point is \(X\text{ J}\). The speed of \(P\) at its lowest point is \(4\text{ m s}^{-1}\). Find \(X\). [3]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]

\includegraphics{figure_2} Two particles \(A\) and \(B\) have masses \(m\) kg and 0.1 kg respectively, where \(m > 0.1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.9 m above horizontal ground (see diagram). The system is released from rest, and while both particles are in motion the tension in the string is 1.5 N. Particle \(B\) does not reach the pulley.

1. Find \(m\). [4]
2. Find the speed at which \(A\) reaches the ground. [2]

\includegraphics{figure_7} A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.

1. It is given that the sloping section of the slide is smooth.
   1. Find the speed of the child when she reaches the bottom of the sloping section. [3]
   2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest. [2]
2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide. [6]

Two particles \(A\) and \(B\), of masses \(2.4\text{kg}\) and \(1.2\text{kg}\) respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at a distance of \(2.1\text{m}\) above a horizontal plane and \(B\) is \(1.5\text{m}\) above the plane. The particles hang vertically and are released from rest. In the subsequent motion \(A\) reaches the plane and does not rebound and \(B\) does not reach the pulley.

1. Show that the tension in the string before \(A\) reaches the plane is \(16\text{N}\) and find the magnitude of the acceleration of the particles before \(A\) reaches the plane. [4]
2. Find the greatest height of \(B\) above the plane. [3]

A particle \(P\) of mass \(0.4\) kg is projected vertically upwards from horizontal ground with speed \(10\) m s\(^{-1}\).

1. Find the greatest height above the ground reached by \(P\). [2]

When \(P\) reaches the ground again, it bounces vertically upwards. At the first instant that it hits the ground, \(P\) loses \(7.2\) J of energy.

1. Find the time between the first and second instants at which \(P\) hits the ground. [4]

\includegraphics{figure_7} Two particles \(P\) and \(Q\) of masses 2.5 kg and 0.5 kg respectively are connected by a light inextensible string that passes over a small smooth pulley fixed at the top of a plane inclined at an angle of \(30°\) to the horizontal. Particle \(P\) is on the plane and \(Q\) hangs below the pulley such that the level of \(Q\) is 2 m below the level of \(P\) (see diagram). Particle \(P\) is released from rest with the string taut and slides down the plane. The plane is rough with coefficient of friction 0.2 between the plane and \(P\).

1. Find the acceleration of \(P\). [5]
2. Use an energy method to find the speed of the particles at the instant when they are at the same vertical height. [5]

\includegraphics{figure_7} The diagram shows a smooth track which lies in a vertical plane. The section \(AB\) is a quarter circle of radius 1.8 m with centre \(O\). The section \(BC\) is a horizontal straight line of length 7.0 m and \(OB\) is perpendicular to \(BC\). The section \(CFE\) is a straight line inclined at an angle of \(\theta°\) above the horizontal. A particle \(P\) of mass 0.5 kg is released from rest at \(A\). Particle \(P\) collides with a particle \(Q\) of mass 0.1 kg which is at rest at \(B\). Immediately after the collision, the speed of \(P\) is \(4\,\text{m}\,\text{s}^{-1}\) in the direction \(BC\). You should assume that \(P\) is moving horizontally when it collides with \(Q\).

1. Show that the speed of \(Q\) immediately after the collision is \(10\,\text{m}\,\text{s}^{-1}\). [4] When \(Q\) reaches \(C\), it collides with a particle \(R\) of mass 0.4 kg which is at rest at \(C\). The two particles coalesce. The combined particle comes instantaneously to rest at \(F\). You should assume that there is no instantaneous change in speed as the combined particle leaves \(C\), nor when it passes through \(C\) again as it returns down the slope.
2. Given that the distance \(CF\) is 0.4 m, find the value of \(\theta\). [4]
3. Find the distance from \(B\) at which \(P\) collides with the combined particle. [5]

A car of mass 1800 kg is towing a trailer of mass 300 kg up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800 N acting on the car and a resistance force of \(F\) N acting on the trailer. The driving force of the car's engine is 3000 N.

1. It is given that \(F = 100\). Find the acceleration of the car and the tension in the tow-bar. [5]
2. It is given instead that the total work done against \(F\) in moving a distance of 50 m up the road is 6000 J. The speed of the car at the start of the 50 m is \(20\) m s\(^{-1}\). Use an energy method to find the speed of the car at the end of the 50 m. [5]

\includegraphics{figure_5} Two particles of masses 0.8 kg and 0.2 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The system is released from rest with both particles 0.5 m above a horizontal floor (see diagram). In the subsequent motion the 0.2 kg particle does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(6 \text{ m s}^{-2}\) and find the tension in the string. [4]
2. When the 0.8 kg particle reaches the floor it comes to rest. Find the greatest height of the 0.2 kg particle above the floor. [3]

A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at an angle of \(\sin^{-1} 0.08\) to the horizontal. The resistances to the motion of the car and trailer are 400 N and 200 N respectively. The car and trailer are connected by a light rigid tow-bar. The car and trailer have speed \(30 \text{ m s}^{-1}\) at the bottom of the hill and \(20 \text{ m s}^{-1}\) at the top of the hill.

1. Use an energy method to find the constant driving force as the car and trailer travel up the hill. [5]
2. After reaching the top of the hill the system consisting of the car and trailer travels along a straight level road. The driving force of the car's engine is 2400 N and the resistances to motion are unchanged. Find the acceleration of the system and the tension in the tow-bar. [4]

\includegraphics{figure_7} Three points \(A\), \(B\) and \(C\) lie on a line of greatest slope of a plane inclined at an angle of \(30°\) to the horizontal, with \(AB = 1\) m and \(BC = 1\) m, as shown in the diagram. A particle of mass 0.2 kg is released from rest at \(A\) and slides down the plane. The part of the plane from \(A\) to \(B\) is smooth. The part of the plane from \(B\) to \(C\) is rough, with coefficient of friction \(\mu\) between the plane and the particle.

1. Given that \(\mu = \frac{1}{2}\sqrt{3}\), find the speed of the particle at \(C\). [8]
2. Given instead that the particle comes to rest at \(C\), find the exact value of \(\mu\). [4]