6.02j Conservation with elastics: springs and strings

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform disc, of mass \(4m\) and radius \(a\), and a uniform ring, of mass \(m\) and radius \(2a\), each have centre \(O\). A wheel is made by fixing three uniform rods, \(OA\), \(OB\) and \(OC\), each of mass \(m\) and length \(2a\), to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through \(A\), perpendicular to the plane of the wheel, is \(42ma^2\). [5] The axis through \(A\) is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with \(O\) above the level of \(A\) and \(AO\) making an angle of \(30°\) with the horizontal. Find the angular speed of the wheel when \(AO\) is horizontal. [3] When \(AO\) is horizontal the disc becomes detached from the wheel. Find the angle that \(AO\) makes with the horizontal when the wheel first comes to instantaneous rest. [6] **OR** The continuous random variable \(T\) has probability density function given by $$f(t) = \begin{cases} 0 & t < 2, \\ \frac{2}{(t-1)^3} & t \geqslant 2. \end{cases}$$

1. Find the distribution function of \(T\), and find also P\((T > 5)\). [3]
2. Consecutive independent observations of \(T\) are made until the first observation that exceeds \(5\) is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding \(5\). Find P\((N > E(N))\). [3]
3. Find the probability density function of \(Y\), where \(Y = \frac{1}{T-1}\). [8]

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos \theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

\(AB\) is a diameter of a uniform circular disc \(D\) of mass \(9m\), radius \(3a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112ma^2\). [5] A particle of mass \(3m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k\sqrt{(ga)}\). Find the value of \(k\), correct to 3 significant figures. [7]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{3l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos\theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

\(AB\) is a diameter of a uniform circular disc \(D\) of mass \(9m\), radius \(3a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112ma^2\). [5] A particle of mass \(3m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k\sqrt{(ga)}\). Find the value of \(k\), correct to 3 significant figures. [7]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{5l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

Day	1	2	3	4	5	6	7	8
\(x\)	1.2	1.4	0.9	1.1	0.8	1.0	0.6	1.5
\(y\)	0.3	0.4	0.6	0.6	0.25	0.75	0.6	0.35

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5\% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5\% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

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]

\(A\) 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]

A uniform rod \(AB\) of length \(4a\) and weight \(W\) is smoothly hinged to a vertical wall at the end \(A\). The rod is held at an angle \(\theta\) above the horizontal by a light elastic string. One end of the string is attached to the point \(C\) on the rod, where \(AC = 3a\). The other end of the string is attached to a point \(D\) on the wall, with \(D\) vertically above \(A\) and such that angle \(ACD = 2\theta\). A particle of weight \(\frac{1}{4}W\) is attached to the rod at \(B\). It is given that \(\tan \theta = \frac{5}{12}\).

1. Show that the tension in the string is \(\frac{17}{12}W\). [4]
2. Find the magnitude and direction of the reaction at the hinge. [5]
3. Given that the natural length of the string is \(2a\), find its modulus of elasticity. [2]

A car of mass \(900\text{kg}\) is moving up a hill inclined at \(\sin^{-1} 0.12\) to the horizontal. The initial speed of the car is \(11\text{ms}^{-1}\). After \(12\text{s}\), the car has travelled \(150\text{m}\) up the hill and has speed \(16\text{ms}^{-1}\). The engine of the car is working at a constant rate of \(24\text{kW}\).

1. Find the work done against the resistive forces during the \(12\text{s}\). [5]
2. The car then travels along a straight horizontal road. There is a resistance to the motion of the car of \((1520 + 4v)\text{N}\) when the speed of the car is \(v\text{ms}^{-1}\). The car travels at a constant speed with the engine working at a constant rate of \(32\text{kW}\). Find this speed. [3]

A car of mass \(1200\) kg is travelling along a straight horizontal road. The power of the car's engine is constant and is equal to \(16\) kW. There is a constant resistance to motion of magnitude \(500\) N.

1. Find the acceleration of the car at an instant when its speed is \(20\) m s\(^{-1}\). [3]
2. Assuming that the power and the resistance forces remain unchanged, find the steady speed at which the car can travel. [2]

The car comes to the bottom of a straight hill of length \(316\) m, inclined at an angle to the horizontal of \(\sin^{-1}(\frac{4}{65})\). The power remains constant at \(16\) kW, but the magnitude of the resistance force is no longer constant and changes such that the work done against the resistance force in ascending the hill is \(128400\) J. The time taken to ascend the hill is \(15\) s.

1. Given that the car is travelling at a speed of \(20\) m s\(^{-1}\) at the bottom of the hill, find its speed at the top of the hill. [6]

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]

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]

\includegraphics{figure_7} The diagram shows the vertical cross-section \(PQR\) of a slide. The part \(PQ\) is a straight line of length \(8\) m inclined at angle \(α\) to the horizontal, where \(\sin α = 0.8\). The straight part \(PQ\) is tangential to the curved part \(QR\) at \(Q\), and \(R\) is \(h\) m above the level of \(P\). The straight part \(PQ\) of the slide is rough and the curved part \(QR\) is smooth. A particle of mass \(0.25\) kg is projected with speed \(15\) m s\(^{-1}\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(PQ\) is \(0.5\).

1. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\). [4]
2. Hence find the speed of the particle at \(Q\). [4]
3. Find the value of \(h\). [3]

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]

\includegraphics{figure_1} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which hangs vertically below \(A\). The particle is also attached to one end of a light elastic string of natural length \(0.25\) m. The other end of this string is attached to a point \(B\) which is \(0.6\) m from \(P\) and on the same horizontal level as \(P\). Equilibrium is maintained by a horizontal force of magnitude \(7\) N applied to \(P\) (see Fig. 1).

1. Calculate the modulus of elasticity of the elastic string. [2]
2. \(P\) is released from rest by removing the \(7\) N force. In its subsequent motion \(P\) first comes to instantaneous rest at a point where \(BP = 0.3\) m and the elastic string makes an angle of \(30°\) with the horizontal (see Fig. 2). \includegraphics{figure_2} Find the value of \(m\). [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_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]

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} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(3a\). The other end of the rod is able to pivot smoothly about the fixed point \(A\). The particle is also attached to one end of a light spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(B\). The points \(A\) and \(B\) are in a horizontal line, a distance \(5a\) apart, and these two points and the rod are in a vertical plane. Initially, \(P\) is held in equilibrium by a vertical force \(F\) with the stretched length of the spring equal to \(4a\) (see diagram). The particle is released from rest in this position and has a speed of \(\frac{6}{5}\sqrt{2ag}\) when the rod becomes horizontal.

1. Find the value of \(k\). [5]
2. Find \(F\) in terms of \(m\) and \(g\). [2]
3. Find, in terms of \(m\) and \(g\), the tension in the rod immediately before it is released. [2]

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]