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

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]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(kmg\), is attached to a fixed point A. The other end of the string is attached to a particle \(P\) of mass \(4m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below A.

1. Show that \(k = \frac{4a}{x-a}\). [1]

An additional particle, of mass \(2m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac{3}{4}a\), its speed is \(\frac{1}{2}\sqrt{ga}\).

1. Find \(x\) in terms of \(a\). [6]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(kmg\), is attached to a fixed point A. The other end of the string is attached to a particle \(P\) of mass \(4m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below A.

1. Show that \(k = \frac{4a}{x-a}\). [1]

An additional particle, of mass \(2m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac{3}{4}a\), its speed is \(\frac{1}{3}\sqrt{ga}\).

1. Find \(x\) in terms of \(a\). [6]

One end of a light elastic string of natural length \(0.8\) m and modulus of elasticity \(36\) N is attached to a fixed point \(O\) on a smooth plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{3}{5}\). A particle \(P\) of mass \(2\) kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of \(O\). The particle is projected with speed \(\sqrt{2}\) m s\(^{-1}\) directly down the plane from the position where \(OP\) is equal to the natural length of the string. Find the maximum extension of the string during the subsequent motion. [5]

\includegraphics{figure_1} A particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\) on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the equilibrium position, the string makes an angle \(\theta\) with the ceiling (see diagram). The string has natural length 0.8 m and modulus of elasticity 50 N.

1. Find the tension in the string. [2]
2. Find the vertical distance between the particle and the ceiling. [3]

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(2mg\). A particle \(Q\) of mass \(km\) is attached to the other end of the string. Particle \(P\) lies on a smooth horizontal table. The string passes through a small smooth hole \(H\) in the table and then passes through a small smooth hole \(H\) in the table. Particle \(P\) moves in a horizontal circle on the surface of the table with constant speed \(\sqrt{\frac{1}{3}ga}\). Particle \(Q\) hangs in equilibrium vertically below the hole with \(HQ = \frac{1}{4}a\).

1. Find, in terms of \(a\), the extension in the string. [4]
2. Find the value of \(k\). [2]

\includegraphics{figure_4} A light spring of natural length \(a\) and modulus of elasticity \(kmg\) is attached to a fixed point \(O\) on a smooth plane inclined to the horizontal at an angle \(\theta\), where \(\sin\theta = \frac{1}{4}\). A particle of mass \(m\) is attached to the lower end of the spring and is held at the point \(A\) on the plane, where \(OA = 2a\) and \(OA\) is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed \(V\) when it passes through the point \(B\) on the plane, where \(OB = \frac{3}{2}a\). The speed of the particle is \(\frac{1}{3}V\) when it passes through the point \(C\) on the plane, where \(OC = \frac{3}{4}a\). Find the value of \(k\). [7]

A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3mg\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. [3]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt{\frac{g}{a}}\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \((k+1)a\).

1. Find the value of \(k\). [4]
2. Find the value of \(\cos\theta\). [2]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3mg\), is attached to a fixed point \(O\) on a smooth horizontal plane. A particle \(P\) of mass \(m\) is attached to the other end of the string and moves in a horizontal circle with centre \(O\). The speed of \(P\) is \(\sqrt{\frac{1}{4}ga}\). Find the extension of the string. [4]

A light elastic string has natural length \(a\) and modulus of elasticity \(12mg\). One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(e > \frac{1}{4}a\). In the subsequent motion the particle has speed \(\sqrt{2ga}\) when it has ascended a distance \(\frac{1}{4}a\). Find \(e\) in terms of \(a\). [6]

\includegraphics{figure_1} A rod \(AB\), of mass \(2m\) and length \(2a\), is suspended from a fixed point \(C\) by two light strings \(AC\) and \(BC\). The rod rests horizontally in equilibrium with \(AC\) making an angle \(\alpha\) with the rod, where \(\tan \alpha = \frac{3}{4}\), and with \(AC\) perpendicular to \(BC\), as shown in Fig. 1.

1. Give a reason why the rod cannot be uniform. [1]
2. Show that the tension in \(BC\) is \(\frac{4}{5}mg\) and find the tension in \(AC\). [5]

The string \(BC\) is elastic, with natural length \(a\) and modulus of elasticity \(kmg\), where \(k\) is constant.

1. Find the value of \(k\). [4]

A light spring of natural length \(L\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The particle is moving vertically. As it passes through the point \(B\) below \(A\), where \(AB = L\), its speed is \(\sqrt{(2gL)}\). The particle comes to instantaneous rest at a point \(C\), \(4L\) below \(A\).

1. Show that the modulus of elasticity of the spring is \(\frac{8mg}{9}\). [4]

At the point \(D\) the tension in the spring is \(mg\).

1. Show that \(P\) performs simple harmonic motion with centre \(D\). [5]
2. Find, in terms of \(L\) and \(g\),
   1. the period of the simple harmonic motion,
   2. the maximum speed of \(P\).
   [5]

\includegraphics{figure_4} A small ball of mass \(3m\) is attached to the ends of two light elastic strings \(AP\) and \(BP\), each of natural length \(l\) and modulus of elasticity \(kmg\). The ends \(A\) and \(B\) of the strings are attached to fixed points on the same horizontal level, with \(AB = 2l\). The mid-point of \(AB\) is \(C\). The ball hangs in equilibrium at a distance \(\frac{3}{4}l\) vertically below \(C\) as shown in Figure 4.

1. Show that \(k = 10\) [7]

The ball is now pulled vertically downwards until it is at a distance \(\frac{15}{8}l\) below \(C\). The ball is released from rest.

1. Find the speed of the ball as it reaches \(C\). [6]

\includegraphics{figure_2} A light horizontal spring, of natural length 0.25 m and modulus of elasticity 52 N, is fastened at one end to a point \(A\). The other end of the spring is fastened to a small wooden block \(B\) of mass 1.5 kg which is on a horizontal table, as shown in Fig. 2. The block is modelled as a particle. The table is initially assumed to be smooth. The block is released from rest when it is a distance 0.3 m from \(A\). By using the principle of the conservation of energy,

1. find, to 3 significant figures, the speed of \(B\) when it is a distance 0.25 m from \(A\). [5]

It is now assumed that the table is rough and the coefficient of friction between \(B\) and the table is 0.6.

1. Find, to 3 significant figures, the minimum distance from \(A\) at which \(B\) can rest in equilibrium. [5]

\includegraphics{figure_5} A small ring \(R\) of mass \(m\) is free to slide on a smooth straight wire which is fixed at an angle of \(30°\) to the horizontal. The ring is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point \(B\), where \(AB = \frac{a}{2}\).

1. Show that \(\lambda = 4mg\). [3]

The ring is pulled down to the point \(C\), where \(BC = \frac{1}{4}a\), and released from rest. At time \(t\) after \(R\) is released the extension of the string is \((\frac{1}{4}a + x)\).

1. Obtain a differential equation for the motion of \(R\) while the string remains taut, and show that it represents simple harmonic motion with period \(\pi\sqrt{\left(\frac{a}{g}\right)}\). [6]
2. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(R\) while the string remains taut. [2]
3. Find, in terms of \(a\) and \(g\), the time taken for \(R\) to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]

A light elastic string \(AB\) of natural length 1.5 m has modulus of elasticity 20 N. The end \(A\) is fixed to a point on a smooth horizontal table. A small ball \(S\) of mass 0.2 kg is attached to the end \(B\). Initially \(S\) is at rest on the table with \(AB = 1.5\) m. The ball \(S\) is then projected horizontally directly away from \(A\) with a speed of 5 m s\(^{-1}\). By modelling \(S\) as a particle,

1. find the speed of \(S\) when \(AS = 2\) m. [5]

When the speed of \(S\) is 1.5 m s\(^{-1}\), the string breaks.

1. Find the tension in the string immediately before the string breaks. [5]

A light elastic string, of natural length \(4a\) and modulus of elasticity \(8mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\).

1. Find the distance \(AO\). [2]

The particle is now pulled down to a point \(C\) vertically below \(O\), where \(OC = d\). It is released from rest. In the subsequent motion the string does not become slack.

1. Show that \(P\) moves with simple harmonic motion of period \(\pi\sqrt{\frac{2a}{g}}\). [7]

The greatest speed of \(P\) during this motion is \(\frac{1}{2}\sqrt{(ga)}\).

1. Find \(d\) in terms of \(a\). [3]

Instead of being pulled down a distance \(d\), the particle is pulled down a distance \(a\). Without further calculation,

1. describe briefly the subsequent motion of \(P\). [2]

A particle \(P\) of mass \(0.8\) kg is attached to one end \(A\) of a light elastic spring \(OA\), of natural length \(60\) cm and modulus of elasticity \(12\) N. The spring is placed on a smooth horizontal table and the end \(O\) is fixed. The particle \(P\) is pulled away from \(O\) to a point \(B\), where \(OB = 85\) cm, and is released from rest.

1. Prove that the motion of \(P\) is simple harmonic with period \(\frac{2\pi}{5}\) s. [5]
2. Find the greatest magnitude of the acceleration of \(P\) during the motion. [2]

Two seconds after being released from rest, \(P\) passes through the point \(C\).

1. Find, to 2 significant figures, the speed of \(P\) as it passes through \(C\). [5]
2. State the direction in which \(P\) is moving 2 s after being released. [1]

Two light elastic strings each have natural length \(0.75\) m and modulus of elasticity \(49\) N. A particle \(P\) of mass \(2\) kg is attached to one end of each string. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 1.5\) m. \includegraphics{figure_2} The particle is held at the mid-point of \(AB\). The particle is released from rest, as shown in Figure 2.

1. Find the speed of \(P\) when it has fallen a distance of \(1\) m. [6]

Given instead that \(P\) hangs in equilibrium vertically below the mid-point of \(AB\), with \(\angle APB = 2\alpha\),

1. show that \(\tan \alpha + 5 \sin \alpha = 5\). [6]

A thin elastic string, of modulus \(\lambda\) N and natural length 20 cm, passes round two small, smooth pegs \(A\) and \(B\) on the same horizontal level to form a closed loop. \(AB = 10\) cm. The ends of the string are attached to a weight \(P\) of mass 0.7 kg. When \(P\) rests in equilibrium, \(APB\) forms an equilateral triangle. \includegraphics{figure_2}

1. Find the value of \(\lambda\). [6 marks]
2. State one assumption that you have made about the weight \(P\), explaining how you have used this assumption in your solution. [1 mark]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(\lambda\) N. The other end of the string is attached to a fixed point \(O\). \(P\) is released from rest at \(O\) and falls vertically downwards under gravity. The greatest distance below \(O\) reached by \(P\) is \(2l\) m.

1. Show that \(\lambda = 4mg\). [3 marks]
2. Find, in terms of \(l\) and \(g\), the speed with which \(P\) passes through the point \(A\), where \(OA = \frac{5l}{4}\) m. [6 marks]

A light elastic string, of natural length \(l\) m and modulus of elasticity \(\frac{mg}{2}\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m\) kg, is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(OE = (l + e)\) m

1. Find the numerical value of the ratio \(e : l\). [2 marks]

\(P\) is now pulled down a further distance \(\frac{3l}{2}\) m from \(E\) and is released from rest. In the subsequent motion, the string remains taut. At time \(t\) s after being released, \(P\) is at a distance \(x\) m below \(E\).

1. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic. [4 marks]
2. Write down the period of the motion. [2 marks]
3. Find the speed with which \(P\) first passes through \(E\) again. [2 marks]
4. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where \(AE = \frac{3l}{4}\) m, is \(\frac{2\pi}{3}\sqrt{\frac{2l}{g}}\) s. [5 marks]

The diagram shows a particle of mass \(0.7\) kg resting on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.25\). A light elastic string, of natural length \(50\) cm and modulus of elasticity \(6.86\) N, is attached to the particle. The string is kept at an angle of \(60°\) to the horizontal and is gradually extended by pulling on it until the particle moves. Show that the particle starts to move when the extension in the string is \(17\) cm. \includegraphics{figure_2} [8 marks]

One end of a light elastic string, of natural length \(3l\) m, is attached to a fixed point \(O\). A particle of mass \(m\) kg is attached to the other end of the string. When the particle hangs freely in equilibrium, the string is extended by a length of \(l\) m. The particle is then pulled down through a further distance \(2l\) m and released from rest.

1. Prove that as long as the string is taut, the particle performs simple harmonic motion about its equilibrium position. [5 marks]
2. Show that the time between the release of the particle and the instant when the string becomes slack is \(\frac{2\pi}{3}\sqrt{\frac{l}{g}}\) s. [4 marks]
3. Find the greatest height reached by the particle above its point of release. [4 marks]
4. Show that the time \(T\) s taken to reach this greatest height from the moment of release is given by \(T = \left(\frac{2\pi}{3} + \sqrt{3}\right)\sqrt{\frac{l}{g}}\). [4 marks]