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

3 A light elastic string has natural length 1.2 m and stiffness \(637 \mathrm { Nm } ^ { - 1 }\).

1. The string is stretched to a length of 1.3 m . Find the tension in the string and the elastic energy stored in the string. One end of this string is attached to a fixed point \(A\). The other end is attached to a heavy ring \(R\) which is free to move along a smooth vertical wire. The shortest distance from A to the wire is 1.2 m (see Fig. 3). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-4_357_337_669_863} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The ring is in equilibrium when the length of the string \(A R\) is 1.3 m .
2. Show that the mass of the ring is 2.5 kg . The ring is given an initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from its equilibrium position. It first comes to rest, instantaneously, in the position where the length of AR is 1.5 m .
3. Find \(u\).
4. Determine whether the ring will rise above the level of A .

1

1. Two light elastic strings, each having natural length 2.15 m and stiffness \(70 \mathrm {~N} \mathrm {~m} ^ { - 1 }\), are attached to a particle P of mass 4.8 kg . The other ends of the strings are attached to fixed points A and B , which are 1.4 m apart at the same horizontal level. The particle P is placed 2.4 m vertically below the midpoint of AB , as shown in Fig. 1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-2_677_474_482_877} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
   1. Show that P is in equilibrium in this position.
   2. Find the energy stored in the string AP . Starting in this equilibrium position, P is set in motion with initial velocity \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards. You are given that P first comes to instantaneous rest at a point C where the strings are slack.
   3. Find the vertical height of C above the initial position of P .
2. 1. Write down the dimensions of force and stiffness (of a spring). A particle of mass \(m\) is performing oscillations with amplitude \(a\) on the end of a spring with stiffness \(k\). The maximum speed \(v\) of the particle is given by \(v = c m ^ { \alpha } k ^ { \beta } a ^ { \gamma }\), where \(c\) is a dimensionless constant.
   2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).

3 Fixed points A and B are 4.8 m apart on the same horizontal level. The midpoint of AB is M . A light elastic string, with natural length 3.9 m and modulus of elasticity 573.3 N , has one end attached to A and the other end attached to \(\mathbf { B }\).

1. Find the elastic energy stored in the string. A particle P is attached to the midpoint of the string, and is released from rest at M . It comes instantaneously to rest when P is 1.8 m vertically below M .
2. Show that the mass of P is 15 kg .
3. Verify that P can rest in equilibrium when it is 1.0 m vertically below M . In general, a light elastic string, with natural length \(a\) and modulus of elasticity \(\lambda\), has its ends attached to fixed points which are a distance \(d\) apart on the same horizontal level. A particle of mass \(m\) is attached to the midpoint of the string, and in the equilibrium position each half of the string has length \(h\), as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-4_280_755_1064_696} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} When the particle makes small oscillations in a vertical line, the period of oscillation is given by the formula $$\sqrt { \frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } } } m ^ { \alpha } a ^ { \beta } \lambda ^ { \gamma }$$
4. Show that \(\frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } }\) is dimensionless.
5. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
6. Hence find the period when the particle P makes small oscillations in a vertical line centred on the position of equilibrium given in part (iii).

2. A particle \(P\) is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point \(A\). \begin{figure}[h] \end{figure} When a horizontal force of magnitude 20 N is applied to \(P\), it rests in equilibrium with the string making an angle of \(30 ^ { \circ }\) with the vertical and \(A P = 1.2 \mathrm {~m}\) as shown in Figure 1.

1. Find the tension in the string.
2. Find the elastic potential energy stored in the string.

1. A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is attached to a fixed point \(A\) and a particle of mass \(m\) is attached to the other end.

The particle is released from rest at \(A\) and falls vertically until it comes to rest instantaneously at the point \(B\). Find the distance \(A B\) in terms of \(a\).
(7 marks)

6. \begin{figure}[h] \end{figure} Figure 2 shows a particle \(P\) of mass \(m\) which lies on a smooth horizontal table. It is attached to a point \(A\) on the table by a light elastic spring of natural length \(3 a\) and modulus of elasticity \(\lambda\), and to a point \(B\) on the table by a light elastic spring of natural length \(2 a\) and modulus of elasticity \(2 \lambda\). The distance between the points \(A\) and \(B\) is \(7 a\).

1. Show that in equilibrium \(A P = \frac { 9 } { 2 } a\). The particle is released from rest at a point \(Q\) where \(Q\) lies on the line \(A B\) and \(A Q = 5 a\).
2. Prove that the subsequent motion of the particle is simple harmonic with a period of \(\pi \sqrt { \frac { 3 m a } { \lambda } }\).
   (9 marks)

1. The mechanism for releasing the ball on a pinball machine contains a light elastic spring of natural length 15 cm and modulus of elasticity \(\lambda\).

The spring is held compressed to a length of 9 cm by a force of 4.5 N .

1. Find \(\lambda\).
2. Find the work done in compressing the spring from a length of 9 cm to a length of 5 cm .
   (4 marks)
   

5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h] \end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.

1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
2. Find the mass of the object.
3. Find the elastic potential energy of the string when the object is suspended from it.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\). \(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),

1. show that the work done against friction is 10.78 J ,
2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
3. find \(\lambda\).

4. \begin{figure}[h] \end{figure} Four identical uniform rods, each of mass \(m\) and length \(2 a\), are freely jointed to form a rhombus \(A B C D\). The rhombus is suspended from \(A\) and is prevented from collapsing by an elastic string which joins \(A\) to \(C\), with \(\angle B A D = 2 \theta , 0 \leq \theta \leq \frac { 1 } { 3 } \pi\), as shown in Fig. 2. The natural length of the elastic string is \(2 a\) and its modulus of elasticity is \(4 m g\).

1. Show that the potential energy, \(V\), of the system is given by $$V = 4 m g a \left[ ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \cos \theta \right] + \text { constant } .$$
2. Hence find the non-zero value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to the mid-point of a light elastic spring of natural length 2 m and modulus of elasticity 4 N . One end \(A\) of the elastic spring is attached to a fixed point on a smooth horizontal table. The spring is then stretched until its length is 4 m and its other end \(B\) is held at a point on the table where \(A B = 4 \mathrm {~m}\). At time \(t = 0 , P\) is at rest on the table at the point \(O\) where \(A O = 2 \mathrm {~m}\), as shown in Fig. 3. The end \(B\) is now moved on the table in such a way that \(A O B\) remains a straight line. At time \(t\) seconds, \(A B = \left( 4 + \frac { 1 } { 2 } \sin 4 t \right) \mathrm { m }\) and \(A P = ( 2 + x ) \mathrm { m }\).

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 x = \sin 4 t$$
2. Hence find the time when \(P\) first comes to instantaneous rest. END

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(8 m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 5 } \mathrm { mg }\), is fixed to \(B\). The other end of the string is attached to a small ring which is free to slide on a smooth straight horizontal wire which is fixed in the same vertical plane as \(A B\) at a height 7a vertically above \(A\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 2.

1. Show that the potential energy \(V\) of the system is given by $$V = \frac { 8 } { 5 } m g a \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant. }$$
2. Hence find the values of \(\theta , 0 \leq \theta \leq \pi\), for which the system is in equilibrium.
3. Determine the nature of these positions of equilibrium.

5. A light elastic string, of natural length \(2 a\) and modulus of elasticity \(m g\), has a particle \(P\) of mass \(m\) attached to its mid-point. One end of the string is attached to a fixed point \(A\) and the other end is attached to a fixed point \(B\) which is at a distance \(4 a\) vertically below \(A\).

1. Show that \(P\) hangs in equilibrium at the point \(E\) where \(A E = \frac { 5 } { 2 } a\). The particle \(P\) is held at a distance \(3 a\) vertically below \(A\) and is released from rest at time \(t = 0\). When the speed of the particle is \(v\), there is a resistance to motion of magnitude \(2 m k v\), where \(k = \sqrt { } \left( \frac { g } { a } \right)\). At time \(t\) the particle is at a distance \(\left( \frac { 5 } { 2 } a + x \right)\) from \(A\).
2. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 k ^ { 2 } x = 0$$
3. Hence find \(x\) in terms of \(t\).

4. \begin{figure}[h] \end{figure} A uniform rod \(P Q\), of length \(2 a\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through the end \(P\). The end \(Q\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { m g } { 2 \sqrt { 3 } }\). The other end of the string is attached to a fixed point \(O\), where \(O P\) is horizontal and \(O P = 2 a\), as shown in Fig. 2. \(\angle O P Q\) is denoted by \(2 \theta\).

1. Show that, when the string is taut, the potential energy of the system is $$- \frac { m g a } { \sqrt { 3 } } ( 2 \cos 2 \theta + \sqrt { 3 } \sin 2 \theta + 2 \sin \theta ) + \text { constant } .$$
2. Verify that there is a position of equilibrium at \(\theta = \frac { \pi } { 6 }\).
3. Determine whether this is a position of stable equilibrium.

5. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a k ^ { 2 }\), where \(k\) is a positive constant. The other end of the string is attached to a fixed point \(A\). At time \(t = 0 , P\) is released from rest from a point which is a distance \(2 a\) vertically below \(A\). When \(P\) is moving with speed \(v\), the air resistance has magnitude \(2 m k v\). At time \(t\), the extension of the string is \(x\).

1. Show that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 k ^ { 2 } x = g$$ You are given that the general solution of this differential equation is $$x = \mathrm { e } ^ { - k t } ( C \sin k t + D \cos k t ) + \frac { g } { 2 k ^ { 2 } } , \quad \text { where } C \text { and } D \text { are constants. }$$
2. Find the value of \(C\) and the value of \(D\). Assuming that the string remains taut,
3. find the value of \(t\) when \(P\) first comes to rest,
4. show that \(2 k ^ { 2 } a < g \left( 1 + \mathrm { e } ^ { \pi } \right)\).

5. A light elastic spring has natural length \(l\) and modulus of elasticity \(m g\). One end of the spring is fixed to a point \(O\) on a rough horizontal table. The other end is attached to a particle \(P\) of mass \(m\) which is at rest on the table with \(O P = l\). At time \(t = 0\) the particle is projected with speed \(\sqrt { } ( g l )\) along the table in the direction \(O P\). At time \(t\) the displacement of \(P\) from its initial position is \(x\) and its speed is \(v\). The motion of \(P\) is subject to air resistance of magnitude \(2 m v \omega\), where \(\omega = \sqrt { \frac { g } { l } }\). The coefficient of friction between \(P\) and the table is 0.5 .

1. Show that, until \(P\) first comes to rest, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + \omega ^ { 2 } x = - 0.5 g$$
2. Find \(x\) in terms of \(t , l\) and \(\omega\).
3. Hence find, in terms of \(\omega\), the time taken for \(P\) to first come to instantaneous rest.
   (3)

7. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(k M\) where \(k\) is a constant, is free to rotate in a vertical plane about the fixed point \(A\). One end of a light inextensible string of length \(6 a\) is attached to the end \(B\) of the rod and passes over a small smooth pulley which is fixed at the point \(P\). The line \(A P\) is horizontal and of length \(2 a\). The other end of the string is attached to a particle of mass \(M\) which hangs vertically below the point \(P\), as shown in Figure 3. The angle \(P A B\) is \(2 \theta\), where \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).

1. Show that the potential energy of the system is $$M g a ( 4 \sin \theta - k \sin 2 \theta ) + \text { constant. }$$ The system has a position of equilibrium when \(\cos \theta = \frac { 3 } { 4 }\).
2. Find the value of \(k\).
3. Hence find the value of \(\cos \theta\) at the other position of equilibrium.
4. Determine the stability of each of the two positions of equilibrium.

4. \begin{figure}[h] \end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.

1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
3. Determine the stability of the position of equilibrium.
   \section*{June 2009}

6. A light elastic spring \(A B\) has natural length \(2 a\) and modulus of elasticity \(2 m n ^ { 2 } a\), where \(n\) is a constant. A particle \(P\) of mass \(m\) is attached to the end \(A\) of the spring. At time \(t = 0\), the spring, with \(P\) attached, lies at rest and unstretched on a smooth horizontal plane. The other end \(B\) of the spring is then pulled along the plane in the direction \(A B\) with constant acceleration \(f\). At time \(t\) the extension of the spring is \(x\).

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
2. Find \(x\) in terms of \(n , f\) and \(t\). Hence find
3. the maximum extension of the spring,
4. the speed of \(P\) when the spring first reaches its maximum extension.
   \section*{June 2009}

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 1.5 kg is attached to the midpoint of a light elastic spring \(A B\), of natural length 2 m and modulus of elasticity 12 N . The end \(A\) of the spring is attached to a fixed point on a smooth horizontal floor. The end \(B\) is held at a point on the floor where \(A B = 6 \mathrm {~m}\). At time \(t = 0 , P\) is at rest on the floor at the point \(O\), where \(A O = 3 \mathrm {~m}\), as shown in Figure 2. The end \(B\) is now moved along the floor in such a way that \(A O B\) remains a straight line and at time \(t\) seconds, \(t \geqslant 0\), $$A B = \left( 6 + \frac { 1 } { 4 } \sin 2 t \right) \mathrm { m }$$ At time \(t\) seconds, \(A P = ( 3 + x ) \mathrm { m }\).

1. Show that, for \(t \geqslant 0\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 16 x = 2 \sin 2 t$$ The general solution of this differential equation is $$x = C \cos 4 t + D \sin 4 t + \frac { 1 } { 6 } \sin 2 t$$ where \(C\) and \(D\) are constants.
2. Find the time at which \(P\) first comes to instantaneous rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{44066c44-e366-4f87-b1b2-c5a894e407fa-20_705_1104_116_420} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}

5. A toy car of mass 0.5 kg is attached to one end \(A\) of a light elastic string \(A B\), of natural length 1.5 m and modulus of elasticity 27 N . Initially the car is at rest on a smooth horizontal floor and the string lies in a straight line with \(A B = 1.5 \mathrm {~m}\). The end \(B\) is moved in a straight horizontal line directly away from the car, with constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds after \(B\) starts to move, the extension of the string is \(x\) metres and the car has moved a distance \(y\) metres. The effect of air resistance on the car can be ignored. By modelling the car as a particle, show that, while the string remains taut,

1. 1. \(x + y = u t\)
   2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 36 x = 0\)
2. Hence show that the string becomes slack when \(t = \frac { \pi } { 6 }\)
3. Find, in terms of \(u\), the speed of the car when \(t = \frac { \pi } { 12 }\)
4. Find, in terms of \(u\), the distance the car has travelled when it first reaches end \(B\) of the string.
   

6. A particle \(P\) of mass 0.2 kg is suspended from a fixed point by a light elastic spring. The spring has natural length 0.8 m and modulus of elasticity 7 N . At time \(t = 0\) the particle is released from rest from a point 0.2 metres vertically below its equilibrium position. The motion of \(P\) is resisted by a force of magnitude \(2 v\) newtons, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\). At time \(t\) seconds, \(P\) is \(x\) metres below its equilibrium position.

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 43.75 x = 0\)
2. Find \(x\) in terms of \(t\).
3. Find the value of \(t\) when \(P\) first comes to instantaneous rest.

7. \begin{figure}[h] \end{figure} Figure 2 shows four uniform rods, each of mass \(m\) and length \(2 a\). The rods are freely hinged at their ends to form a rhombus \(A B C D\). Point \(A\) is attached to a fixed point on a ceiling and the rhombus hangs freely with \(C\) vertically below \(A\). A light elastic spring of natural length \(2 a\) and modulus of elasticity \(7 m g\) connects the points \(A\) and \(C\). A particle of mass \(3 m\) is attached to point \(C\).

1. Show that, when \(A D\) is at an angle \(\theta\) to the downward vertical, the potential energy \(V\) of the system is given by $$V = 28 m g a \cos ^ { 2 } \theta - 48 m g a \cos \theta + \text { constant }$$ Given that \(\theta > 0\)
2. find the value of \(\theta\) for which the system is in equilibrium,
3. determine the stability of this position of equilibrium.

5. A particle \(P\) of mass \(m\) is fixed to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a n ^ { 2 }\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at a point which is a distance \(2 a\) vertically below \(O\). The air resistance is modelled as having magnitude \(2 m n v\), where \(v\) is the speed of \(P\).

1. Show that, when the extension of the string is \(x\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 n \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 n ^ { 2 } x = g$$
2. Find \(x\) in terms of \(t\).

4 \includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-2_439_1045_1512_550} Two small smooth pegs \(A\) and \(B\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(C\) is the mid-point of \(A B\). A uniform rod \(C D\), of mass \(m\) and length \(a\), is freely pivoted at \(C\) and can rotate in the vertical plane containing \(A B\), with \(D\) below the level of \(A B\). A light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), passes round the peg \(A\) and its ends are attached to \(C\) and \(D\). Another light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), passes round the peg \(B\) and its ends are also attached to \(C\) and \(D\). The angle \(C A D\) is \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), so that the angle \(B C D\) is \(2 \theta\) (see diagram).

1. Taking \(A B\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$\frac { 1 } { 2 } m g a ( 14 - 2 \cos 2 \theta - \sin 2 \theta )$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.