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

A light elastic spring, of natural length \(5a\) and modulus of elasticity \(10mg\), has one end attached to a fixed point \(A\) on a ceiling. A particle \(P\) of mass \(2m\) is attached to the other end of the spring and \(P\) hangs freely in equilibrium at the point \(O\).

1. Find the distance \(AO\). [3]

The particle is now pulled vertically downwards a distance \(\frac{1}{2}a\) from \(O\) and released from rest.

1. Show that \(P\) moves with simple harmonic motion. [4]
2. Find the period of the motion. [2]

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{2}{5}\). The particle is held at a point \(A\) on the plane, where \(OA = \frac{5}{4}l\), and released from rest. The particle comes to rest at the point \(B\).

1. Show that \(OB < l\) [4]
2. Find the distance \(OB\). [3]

\includegraphics{figure_2} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The coefficient of friction between the particle and the plane is 0.15. The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\). [9]

A light elastic string, of natural length \(3a\) and modulus of elasticity \(6mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(2m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\), vertically below \(A\).

1. Find the distance \(AO\). [3]

The particle is now raised to point \(C\) vertically below \(A\), where \(AC > 3a\), and is released from rest.

1. Show that \(P\) moves with simple harmonic motion of period \(2\pi\sqrt{\frac{a}{g}}\). [5]

It is given that \(OC = \frac{1}{4}a\).

1. Find the greatest speed of \(P\) during the motion. [3]

The point \(D\) is vertically above \(O\) and \(OD = \frac{1}{8}a\). The string is cut as \(P\) passes through \(D\), moving upwards.

1. Find the greatest height of \(P\) above \(O\) in the subsequent motion. [4]

One end of a light elastic string, of natural length 2 m and modulus of elasticity 19.6 N, is attached to a fixed point \(A\). A small ball \(B\) of mass 0.5 kg is attached to the other end of the string. The ball is released from rest at \(A\) and first comes to instantaneous rest at the point \(C\), vertically below \(A\).

1. Find the distance \(AC\). [6]
2. Find the instantaneous acceleration of \(B\) at \(C\). [3]

\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 light elastic string has natural length \(8\) m and modulus of elasticity \(80\) N. The ends of the string are attached to fixed points \(P\) and \(Q\) which are on the same horizontal level and \(12\) m apart. A particle is attached to the mid-point of the string and hangs in equilibrium at a point \(4.5\) m below \(PQ\).

1. Calculate the weight of the particle. [6]
2. Calculate the elastic energy in the string when the particle is in this position. [3]

\includegraphics{figure_4} \(A\) and \(B\) are two points on a smooth horizontal floor, where \(AB = 5\) m. A particle \(P\) has mass \(0.5\) kg. One end of a light elastic spring, of natural length \(2\) m and modulus of elasticity \(16\) N, is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length \(1\) m and modulus of elasticity \(12\) N, are attached to \(P\) and \(B\), as shown in Figure 4.

1. Find the extensions in the two springs when the particle is at rest in equilibrium. [5]

Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).

1. Show that \(P\) oscillates with simple harmonic motion about the equilibrium position. [4]
2. Given that the initial speed of \(P\) is \(\sqrt{10}\) m s\(^{-1}\), find the proportion of time in each complete oscillation for which \(P\) stays within \(0.25\) m of the equilibrium position. [7]

An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20°\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg. The coefficient of friction between the particle and the plane is 0.8. The three points, \(A\), \(B\) and \(C\), lie on a line of greatest slope. The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane. \includegraphics{figure_8}

1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N, correct to three significant figures. [3 marks]
2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
   1. Find \(x\). [8 marks]
   2. Find the acceleration of the particle as it starts to move back towards \(A\). [4 marks]

A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres. The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane. \includegraphics{figure_8} The particle is attached to two light elastic strings, \(PQ\) and \(PR\). One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N. The particle is released from rest at the point \(A\). The coefficient of friction between the particle and the plane is \(0.4\). Find the distance of the particle from \(Q\) when it is next at rest. [8 marks]

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]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length 0·5 m and modulus of elasticity \(\frac{mg}{2}\) N. The other end of the string is attached to a fixed point \(O\) and \(P\) hangs vertically below \(O\).

1. Find the stretched length of the string when \(P\) rests in equilibrium. [3 marks]
2. Find the elastic potential energy stored in the string in the equilibrium position. [2 marks]

\(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.

1. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position. [2 marks]
2. Explain the discrepancy between your answers to parts (b) and (c). [1 mark]

A particle \(P\), of mass \(m\) kg, is attached to two light elastic strings, each of natural length \(l\) m and modulus of elasticity \(3mg\) N. The other ends of the strings are attached to the fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 2l\) m. \includegraphics{figure_4} If \(P\) rests in equilibrium vertically below the mid-point of \(AB\), with each string making an angle \(\theta\) with the vertical, show that $$\cot \theta - \cos \theta = \frac{1}{6}.$$ [8 marks]

A light elastic string, of natural length 0·8 m, has one end fastened to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0·5 kg. When \(P\) hangs in equilibrium, the length of the string is 1·5 m.

1. Calculate the modulus of elasticity of the string. [3 marks]

\(P\) is displaced to a point 0·5 m vertically below its equilibrium position and released from rest. \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Show that the subsequent motion of \(P\) is simple harmonic, with period 1·68 s. [5 marks] \item Calculate the maximum speed of \(P\) during its motion. [3 marks] \item Show that the time taken for \(P\) to first reach a distance 0·25 m from the point of release is 0·28 s, to 2 significant figures. [4 marks] \end{enumerate]

A particle of mass \(m\) kg is attached to the end \(B\) of a light elastic string \(AB\). The string has natural length \(l\) m and modulus of elasticity \(\lambda\) N. \includegraphics{figure_3} The end \(A\) is attached to a fixed point on a smooth plane inclined at an angle \(\alpha\) to the horizontal, as shown, and the particle rests in equilibrium with the length \(AB = \frac{5l}{4}\) m.

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

The particle is now moved and held at rest at \(A\) with the string slack. It is then gently released so that it moves down the plane along a line of greatest slope.

1. Find the greatest distance from \(A\) that the particle reaches down the plane. [6 marks]

A light spring, of natural length 30 cm, is fixed in a vertical position. When a small ball of mass 0.4 kg rests on top of it, the spring is compressed by 10 cm. The ball is then held at a height of 15 cm vertically above the top of the spring and released from rest. Calculate the maximum compression of the string in the resulting motion. [7 marks]

A particle of mass \(m\) kg is attached to one end of an 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 \(O\). The particle hangs in equilibrium at a point \(C\).

1. 1. Prove that if the particle is slightly displaced in a vertical direction, it will perform simple harmonic motion about \(C\). [6 marks]
   2. Find the period, \(T\) s, of the motion in terms of \(l\), \(m\) and \(\lambda\). [1 mark]
   3. Explain the significance of the term 'slightly' as used in (i) above. [1 mark]

When an additional mass \(M\) is attached to the particle, it is found that the system oscillates about a point \(D\), at a distance \(d\) below \(C\), with period \(T_1\) s.

1. 1. Write down an expression for \(T_1\) in terms of \(l\), \(m\), \(M\) and \(\lambda\). [2 marks]
   2. Hence show that \(T_1^2 - T^2 = \frac{4\pi^2 d}{g}\). [5 marks]

A particle \(P\) of mass 0.2 kg moves in a horizontal circle on one end of an elastic string whose other end is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\pi\) rad s\(^{-1}\). The natural length of the string is 1 m and, while \(P\) is in motion, the distance \(OP = 1.15\) m.

1. Calculate, to 3 significant figures, the modulus of elasticity of the string. [6 marks]

The motion now ceases and \(P\) hangs at rest vertically below \(O\).

1. Show that the extension in the string in this position is about 13 cm. [3 marks]

A small stone \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus \(3mg\) N and natural length \(2l\) m. The other end of the string is fixed to a point \(O\) at a height \(3l\) m above a horizontal surface. \(P\) is released from rest at \(O\); it hits the surface and rebounds to a height of \(2l\) m. The coefficient of restitution between \(P\) and the surface is \(e\). Calculate the value of \(e\). [9 marks] State one assumption (other than the string being light) that you have used in your solution. [1 mark]