6.02h Elastic PE: 1/2 k x^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]

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]

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]

\includegraphics{figure_6} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m. The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(AB\) (see diagram).

1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N. [4]

\(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.

1. Find the speed with which \(P\) passes through \(M\). [6]

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]

\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 \(0.15\) 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]

\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. Find the value of \(m\). [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]

\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]

\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]

\includegraphics{figure_2} A uniform rigid rod \(AB\) of length \(1.2\text{ m}\) and weight \(8\text{ N}\) has a particle of weight \(2\text{ N}\) attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(20\text{ N}\) is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed \(0.8\text{ m}\) vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta°\) to the vertical (see diagram).

1. Show that the tension in the string is \(20\sin\theta\text{ N}\). [1]
2. Explain why the part of the string attached to the ring is perpendicular to the rod. [1]
3. Find \(\theta\). [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]

\includegraphics{figure_2} A uniform rigid rod \(AB\) of length \(1.2\,\text{m}\) and weight \(8\,\text{N}\) has a particle of weight \(2\,\text{N}\) attached at the end \(B\). The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string of natural length \(0.8\,\text{m}\) and modulus of elasticity \(20\,\text{N}\) is attached to the hinge. The string passes over a small smooth pulley \(P\) fixed \(0.8\,\text{m}\) vertically above the hinge. The other end of the string is attached to a small light smooth ring \(R\) which can slide on the rod. The system is in equilibrium with the rod inclined at an angle \(\theta°\) to the vertical (see diagram).

1. Show that the tension in the string is \(20\sin\theta\,\text{N}\). [1]
2. Explain why the part of the string attached to the ring is perpendicular to the rod. [1]
3. Find \(\theta\). [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]