6.02g Hooke's law: T = k*x or T = lambda*x/l

A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(mg\) N. The other end of the string is attached to a fixed point \(O\). Given that the string makes an angle of \(60°\) with the vertical,

1. show that \(OP = 3l\) m. [4 marks]
2. Find, in terms of \(l\) and \(g\), the angular speed of \(P\). [4 marks]

Two light elastic strings, each of length \(l\) m and modulus of elasticity \(\lambda\) N, are attached to a particle \(P\) of mass \(m\) kg. The other ends of the strings are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(AB = 2l\) m. \(P\) is held vertically below the mid-point of \(AB\), with each string taut and inclined at \(30°\) to the horizontal, and released from rest. Given that \(P\) comes to instantaneous rest when each string makes an angle of \(60°\) with the horizontal, show that \(\lambda = \frac{3mg}{6 - 2\sqrt{3}}\). \includegraphics{figure_1} [10 marks]

A particle \(P\) of mass \(m\) kg hangs in equilibrium at one end of a light spring, of natural length \(l\) m and modulus of elasticity \(\lambda\) N, whose other end is fixed at a point vertically above \(P\). In this position the length of the spring is \((l + e)\) m. When \(P\) is displaced vertically through a small distance and released, it performs simple harmonic motion with 5 oscillations per second.

1. Show that \(\frac{\lambda}{l} = 100n^2m\). [8 marks]
2. Express \(e\) in terms of \(g\). [2 marks]
3. Determine, in terms of \(m\) and \(l\), the magnitude of the tension in the spring when it is stretched to twice its natural length. [2 marks]

Two particles \(A\) and \(B\), of masses \(M\) kg and \(m\) kg respectively, are connected by a light inextensible string passing over a smooth fixed pulley. \(B\) is placed on a smooth horizontal table and \(A\) hangs freely, as shown. \(B\) is attached to a spring of natural length \(l\) m and modulus of elasticity \(\lambda\) N, whose other end is fixed to a vertical wall. \includegraphics{figure_3} The system starts to move from rest when the string is taut and the spring neither extended nor compressed. \(A\) does not reach the ground, nor does \(B\) reach the pulley, during the motion.

1. Show that the maximum extension of the spring is \(\frac{2Mgl}{\lambda}\) m. [3 marks]
2. If \(M = 3\), \(m = 1.5\) and \(\lambda = 35l\), find the speed of \(A\) when the extension in the spring is \(0.5\) m. [6 marks]

A particle \(P\) of mass \(m\) kg is fixed to one end of a light elastic string of modulus \(mg\) N and natural length \(l\) m. The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(OX = \frac{5l}{4}\) m.

1. Find the coefficient of friction between \(P\) and the table. [2 marks]

\(P\) is now given a small displacement \(x\) m horizontally along \(OX\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.

1. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. [4 marks]

If \(P\) is held at the point where the extension in the string is \(l\) m and then released,

1. show that the string becomes slack after a time \(\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{3}\right)\right)\sqrt{\frac{l}{g}}\) s. [5 marks]
2. Determine the speed of \(P\) when it reaches \(O\). [4 marks]

The figure shows a particle \(P\), of mass 0·8 kg, attached to the ends of two light elastic strings. \(AP\) has natural length 20 cm and modulus of elasticity \(\lambda\) N. \(BP\) has natural length 20 cm and modulus of elasticity \(\mu\) N. \(A\) and \(B\) are fixed to points on the same horizontal level so that \(AB = 50\) cm. When \(P\) is suspended in equilibrium, \(AP = 30\) cm and \(BP = 40\) cm. Calculate the values of \(\lambda\) and \(\mu\). \includegraphics{figure_2} [9 marks]

A light spring of natural length 0.6 metres is compressed to a length of 0.4 metres by a force of 20 newtons. The stiffness of the spring is \(k\) N m\(^{-1}\) Find \(k\) Circle your answer. [1 mark] 20 50 100 200

In this question use \(g\) as \(10\,\text{m}\,\text{s}^{-2}\) A smooth plane is inclined at \(30°\) to the horizontal. The fixed points \(A\) and \(B\) are 3.6 metres apart on the line of greatest slope of the plane, with \(A\) higher than \(B\) A particle \(P\) of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points \(A\) and \(B\) respectively. The particle \(P\) moves on a straight line that passes through \(A\) and \(B\) \includegraphics{figure_2} The natural length of the string \(AP\) is 1.4 metres. When the extension of the string \(AP\) is \(e_A\) metres, the tension in the string \(AP\) is \(7e_A\) newtons. The natural length of the string \(BP\) is 1 metre. When the extension of the string \(BP\) is \(e_B\) metres, the tension in the string \(BP\) is \(9e_B\) newtons. The particle \(P\) is held at the point between \(A\) and \(B\) which is 0.2 metres from its equilibrium position and lower than its equilibrium position. The particle \(P\) is then released from rest. At time \(t\) seconds after \(P\) is released, its displacement towards \(B\) from its equilibrium position is \(x\) metres.

1. Show that during the subsequent motion the object satisfies the equation $$\ddot{x} + 50x = 0$$ Fully justify your answer. [5 marks]
2. The experiment is repeated in a large tank of oil. During the motion the oil causes a resistive force of \(kv\) newtons to act on the particle, where \(v\,\text{m}\,\text{s}^{-1}\) is the speed of the particle. The oil causes critical damping to occur.
   1. Show that \(k = \frac{16\sqrt{2}}{5}\) [3 marks]
   2. Find \(x\) in terms of \(t\), giving your answer in exact form. [6 marks]
   3. Calculate the maximum speed of the particle. [5 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) Two light elastic strings each have one end attached to a small ball \(B\) of mass 0.5 kg The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart with \(A\) vertically above \(C\) The whole system is in a thin tube of oil, as shown in the diagram below. \includegraphics{figure_18} The string connecting \(A\) and \(B\) has natural length 2 metres, and the tension in this string is \(7e\) newtons when the extension is \(e\) metres. The string connecting \(B\) and \(C\) has natural length 3 metres, and the tension in this string is \(3e\) newtons when the extension is \(e\) metres.

1. Find the extension of each string when the system is in equilibrium. [3 marks]
2. It is known that in a large bath of oil, the oil causes a resistive force of magnitude \(4.5v\) newtons to act on the ball, where \(v\) m s\(^{-1}\) is the speed of the ball. Use this model to answer part (b)(i) and part (b)(ii).
   1. The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards \(C\), and released from rest. Show that during the subsequent motion the particle satisfies the differential equation $$\frac{d^2x}{dt^2} + 9\frac{dx}{dt} + 20x = 0$$ where \(x\) metres is the displacement of the particle below the equilibrium position at time \(t\) seconds after the particle is released. [3 marks]
   2. Find \(x\) in terms of \(t\) [5 marks]
3. State one limitation of the model used in part (b) [1 mark]

The fixed points E and F are on the same horizontal level with EF = 1.6 m. A light string has natural length 0.7 m and modulus of elasticity 29.4 N. One end of the string is attached to E and the other end is attached to a particle of mass \(M\) kg. A second string, identical to the first, has one end attached to F and the other end attached to the particle. The system is in equilibrium in a vertical plane with each string stretched to a length of 1 m, as shown in Fig. 3. \includegraphics{figure_3}

1. Find the tension in each string. [2]
2. Find \(M\). [3]

A right circular cone C of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of C. The other end of the string is attached to a particle P of mass 2.5 kg. P moves in a horizontal circle with constant speed and in contact with the smooth curved surface of C. The extension of the string is 1.5 m.

1. Find the tension in the string. [2]
2. Find the speed of P. [7]

3 A light elastic string has natural length 0.8 m and modulus of elasticity 16 N . One end of the string is attached to a fixed point \(O\), and a particle \(P\) of mass 0.4 kg is attached to the other end of the string. The particle \(P\) hangs in equilibrium vertically below \(O\).

1. Show that the extension of the string is 0.2 m . \(P\) is projected vertically downwards from the equilibrium position. \(P\) first comes to instantaneous rest at the point where \(O P = 1.4 \mathrm {~m}\).
2. Calculate the speed at which \(P\) is projected.
3. Find the speed of \(P\) at the first instant when the string subsequently becomes slack.

2 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 45 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at \(O\) and falls vertically. Find the extension of the string when \(P\) is at its lowest position.