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

6 A light elastic string has one end attached to a point \(A\) fixed on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle of mass 6 kg . The elastic string has natural length 4 metres and modulus of elasticity 300 newtons. The particle is pulled down the plane in the direction of the line of greatest slope through \(A\). The particle is released from rest when it is 5.5 metres from \(A\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_314_713_1900_660}

1. Calculate the elastic potential energy of the string when the particle is 5.5 metres from the point \(A\).
2. Show that the speed of the particle when the string becomes slack is \(3.66 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
3. Show that the particle will not reach point \(A\) in the subsequent motion.

3 A light elastic rope has natural length 15 m . One end of the rope is attached to a fixed point O and the other end is attached to a small rock of mass 12 kg . When the rock is hanging in equilibrium vertically below O , the length of the rope is 15.8 m .

1. Show that the modulus of elasticity of the rope is 2205 N . The rock is pulled down to the point 20 m vertically below O , and is released from rest in this position. It moves upwards, and comes to rest instantaneously, with the rope slack, at the point A .
2. Find the acceleration of the rock immediately after it is released.
3. Use an energy method to find the distance OA. At time \(t\) seconds after release, the rope is still taut and the displacement of the rock below the equilibrium position is \(x\) metres.
4. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 12.25 x\).
5. Write down an expression for \(x\) in terms of \(t\), and hence find the time between releasing the rock and the rope becoming slack.

3 Ben has mass 60 kg and he is considering doing a bungee jump using an elastic rope with natural length 32 m . One end of the rope is attached to a fixed point O , and the other end is attached to Ben. When Ben is supported in equilibrium by the rope, the length of the rope is 32.8 m . To predict what will happen, Ben is modelled as a particle B, the rope is assumed to be light, and air resistance is neglected. B is released from rest at O and falls vertically. When the rope becomes stretched, \(x \mathrm {~m}\) denotes the extension of the rope.

1. Find the stiffness of the rope.
2. Use an energy argument to show that, when B comes to rest instantaneously with the rope stretched, $$x ^ { 2 } - 1.6 x - 51.2 = 0$$ Hence find the length of the rope when B is at its lowest point.
3. Show that, while the rope is stretched, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12.25 x = 9.8$$ where \(t\) is the time measured in seconds.
4. Find the time taken for B to travel between the equilibrium position \(( x = 0.8 )\) and the lowest point.
5. Find the acceleration of \(\mathbf { B }\) when it is at the lowest point, and comment on the implications for Ben.

1

1. 1. Write down the dimensions of force and the dimensions of density. When a wire, with natural length \(l _ { 0 }\) and cross-sectional area \(A\), is stretched to a length \(l\), the tension \(F\) in the wire is given by $$F = \frac { E A \left( l - l _ { 0 } \right) } { l _ { 0 } }$$ where \(E\) is Young's modulus for the material from which the wire is made.
   2. Find the dimensions of Young's modulus \(E\). A uniform sphere of radius \(r\) is made from material with density \(\rho\) and Young's modulus \(E\). When the sphere is struck, it vibrates with periodic time \(t\) given by $$t = k r ^ { \alpha } \rho ^ { \beta } E ^ { \gamma }$$ where \(k\) is a dimensionless constant.
   3. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. Fig. 1 shows a fixed point A that is 1.5 m vertically above a point B on a rough horizontal surface. A particle P of mass 5 kg is at rest on the surface at a distance 0.8 m from B , and is connected to A by a light elastic string with natural length 1.5 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-2_405_538_1338_845} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} The coefficient of friction between P and the surface is 0.4 , and P is on the point of sliding. Find the stiffness of the string.

2

1. A small ball of mass 0.01 kg is moving in a vertical circle of radius 0.55 m on the smooth inside surface of a fixed sphere also of radius 0.55 m . When the ball is at the highest point of the circle, the normal reaction between the surface and the ball is 0.1 N . Modelling the ball as a particle and neglecting air resistance, find
   1. the speed of the ball when it is at the highest point of the circle,
   2. the normal reaction between the surface and the ball when the vertical height of the ball above the lowest point of the circle is 0.15 m .
2. A small object Q of mass 0.8 kg moves in a circular path, with centre O and radius \(r\) metres, on a smooth horizontal surface. A light elastic string, with natural length 2 m and modulus of elasticity 160 N , has one end attached to Q and the other end attached to O . The object Q has a constant angular speed of \(\omega\) rad s \(^ { - 1 }\).
   1. Show that \(\omega ^ { 2 } = \frac { 100 ( r - 2 ) } { r }\) and deduce that \(\omega < 10\).
   2. Find expressions, in terms of \(r\) only, for the elastic energy stored in the string, and for the kinetic energy of Q . Show that the kinetic energy of Q is greater than the elastic energy stored in the string.
   3. Given that the angular speed of Q is \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tension in the string.

3 An elastic rope has natural length 25 m and modulus of elasticity 980 N . One end of the rope is attached to a fixed point O , and a rock of mass 5 kg is attached to the other end; the rock is always vertically below O.

1. Find the extension of the rope when the rock is hanging in equilibrium. When the rock is moving with the rope stretched, its displacement is \(x\) metres below the equilibrium position at time \(t\) seconds.
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 7.84 x\). The rock is released from a position where the rope is slack, and when the rope just becomes taut the speed of the rock is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance below the equilibrium position at which the rock first comes instantaneously to rest.
4. Find the maximum speed of the rock.
5. Find the time between the rope becoming taut and the rock first coming to rest.
6. State three modelling assumptions you have made in answering this question.

1

1. 1. Write down the dimensions of density, kinetic energy and power. A sphere of radius \(r\) is moved at constant velocity \(v\) through a fluid.
   2. In a viscous fluid, the power required is \(6 \pi \eta r v ^ { 2 }\), where \(\eta\) is the viscosity of the fluid. Find the dimensions of viscosity.
   3. In a non-viscous fluid, the power required is \(k \rho ^ { \alpha } r ^ { \beta } v ^ { \gamma }\), where \(\rho\) is the density of the fluid and \(k\) is a dimensionless constant. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. A rock of mass 5.5 kg is connected to a fixed point O by a light elastic rope with natural length 1.2 m . The rock is released from rest in a position 2 m vertically below O , and it next comes to instantaneous rest when it is 1.5 m vertically above O . Find the stiffness of the rope.

4 Fig. 4 shows a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Two fixed points A and B on the plane are 4.55 m apart with B higher than A on a line of greatest slope. A particle P of mass 0.25 kg is in contact with the plane and is connected to A and to B by two light elastic strings. The string AP has natural length 1.5 m and modulus of elasticity 7.35 N ; the string BP has natural length 2.5 m and modulus of elasticity 7.35 N . The particle P moves along part of the line AB , with both strings taut throughout the motion. \begin{figure}[h] \end{figure}

1. Show that, when \(\mathrm { AP } = 1.55 \mathrm {~m}\), the acceleration of P is zero.
2. Taking \(\mathrm { AP } = ( 1.55 + x ) \mathrm { m }\), write down the tension in the string AP , in terms of \(x\), and show that the tension in the string BP is \(( 1.47 - 2.94 x ) \mathrm { N }\).
3. Show that the motion of P is simple harmonic, and find its period. The particle P is released from rest with \(\mathrm { AP } = 1.5 \mathrm {~m}\).
4. Find the time after release when P is first moving down the plane with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 A block of mass 200 kg is connected to a horizontal ceiling by four identical light elastic ropes, each having natural length 7 m and stiffness \(180 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). It is also connected to the floor by a single light elastic rope having stiffness \(80 \mathrm { Nm } ^ { - 1 }\). Throughout this question you may assume that all five ropes are stretched and vertical, and you may neglect air resistance. \begin{figure}[h] \end{figure} Fig. 3 shows the block resting in equilibrium, with each of the top ropes having length 10 m and the bottom rope having length 8 m .

1. Find the tension in one of the top ropes.
2. Find the natural length of the bottom rope. The block now moves vertically up and down. At time \(t\) seconds, the block is \(x\) metres below its equilibrium position.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\). The motion is started by pulling the block down 2.2 m below its equilibrium position and releasing it from rest. The block then executes simple harmonic motion with amplitude 2.2 m .
4. Find the maximum magnitude of the acceleration of the block.
5. Find the speed of the block when it has travelled 3.8 m from its starting point.
6. Find the distance travelled by the block in the first 5 s .

3 A bungee jumper of mass 75 kg is connected to a fixed point A by a light elastic rope with stiffness \(300 \mathrm { Nm } ^ { - 1 }\). The jumper starts at rest at A and falls vertically. The lowest point reached by the jumper is 40 m vertically below A. Air resistance may be neglected.

1. Find the natural length of the rope.
2. Show that, when the rope is stretched and its extension is \(x\) metres, \(\ddot { x } + 4 x = 9.8\). The substitution \(y = x - c\), where \(c\) is a constant, transforms this equation to \(\ddot { y } = - 4 y\).
3. Find \(c\), and state the maximum value of \(y\).
4. Using standard simple harmonic motion formulae, or otherwise, find
   (A) the maximum speed of the jumper,
   (B) the maximum deceleration of the jumper.
5. Find the time taken for the jumper to fall from A to the lowest point.

3 Two fixed points X and Y are 14.4 m apart and XY is horizontal. The midpoint of XY is M . A particle P is connected to X and to Y by two light elastic strings. Each string has natural length 6.4 m and modulus of elasticity 728 N . The particle P is in equilibrium when it is 3 m vertically below M, as shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the tension in each string when P is in the equilibrium position.
2. Show that the mass of P is 12.5 kg . The particle P is released from rest at M , and moves in a vertical line.
3. Find the acceleration of P when it is 2.1 m vertically below M .
4. Explain why the maximum speed of P occurs at the equilibrium position.
5. Find the maximum speed of P .

1

1. 1. Find the dimensions of power. In a particle accelerator operating at power \(P\), a charged sphere of radius \(r\) and density \(\rho\) has its speed increased from \(u\) to \(2 u\) over a distance \(x\). A student derives the formula $$x = \frac { 28 \pi r ^ { 3 } u ^ { 2 } \rho } { 9 P }$$
   2. Show that this formula is not dimensionally consistent.
   3. Given that there is only one error in this formula for \(x\), obtain the correct formula.
2. A light elastic string, with natural length 1.6 m and stiffness \(150 \mathrm { Nm } ^ { - 1 }\), is stretched between fixed points A and B which are 2.4 m apart on a smooth horizontal surface.
   1. Find the energy stored in the string. A particle is attached to the mid-point of the string. The particle is given a horizontal velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to AB (see Fig. 1.1), and it comes instantaneously to rest after travelling a distance of 0.9 m (see Fig. 1.2). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_639} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_1128} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   2. Find the mass of the particle.

3 A fixed point A is 12 m vertically above a fixed point B. A light elastic string, with natural length 3 m and modulus of elasticity 1323 N , has one end attached to A and the other end attached to a particle P of mass 15 kg . Another light elastic string, with natural length 4.5 m and modulus of elasticity 1323 N , has one end attached to B and the other end attached to P .

1. Verify that, in the equilibrium position, \(\mathrm { AP } = 5 \mathrm {~m}\). The particle P now moves vertically, with both strings AP and BP remaining taut throughout the motion. The displacement of P above the equilibrium position is denoted by \(x \mathrm {~m}\) (see Fig. 3). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-4_405_360_751_849} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Show that the tension in the string AP is \(441 ( 2 - x ) \mathrm { N }\) and find the tension in the string BP .
3. Show that the motion of P is simple harmonic, and state the period. The minimum length of AP during the motion is 3.5 m .
4. Find the maximum length of AP .
5. Find the speed of P when \(\mathrm { AP } = 4.1 \mathrm {~m}\).
6. Find the time taken for AP to increase from 3.5 m to 4.5 m .

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 .

3 A small block B has mass 2.5 kg . A light elastic string connects B to a fixed point P , and a second light elastic string connects \(B\) to a fixed point \(Q\), which is 6.5 m vertically below \(P\). The string PB has natural length 3.2 m and stiffness \(35 \mathrm { Nm } ^ { - 1 }\); the string BQ has natural length 1.8 m and stiffness \(5 \mathrm { Nm } ^ { - 1 }\). The block B is released from rest in the position 4.4 m vertically below P . You are given that B performs simple harmonic motion along part of the line PQ, and that both strings remain taut throughout the motion. Air resistance may be neglected. At time \(t\) seconds after release, the length of the string PB is \(x\) metres (see Fig. 3). \begin{figure}[h] \end{figure}

1. Find, in terms of \(x\), the tension in the string PB and the tension in the string BQ .
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 64 - 16 x\).
3. Find the value of \(x\) when B is at the centre of oscillation.
4. Find the period of oscillation.
5. Write down the amplitude of the motion and find the maximum speed of B.
6. Find the time after release when \(B\) is first moving downwards with speed \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 In trials for a vehicle emergency stopping system, a small car of mass 400 kg is propelled towards a buffer. The buffer is modelled as a light spring of stiffness \(5000 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). One end of the spring is fixed, and the other end points directly towards the oncoming car. Throughout this question, there is no driving force acting on the car, and there are no resistances to motion apart from those specifically mentioned. At first, the buffer is mounted on a horizontal surface, and the car has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the buffer, as shown in Fig. 2.1. \begin{figure}[h] \end{figure}

1. Find the compression of the spring when the car comes (instantaneously) to rest. The buffer is now mounted on a slope making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). The car is released from rest and travels 7.35 m down the slope before hitting the buffer, as shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-3_268_1091_1329_529} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Verify that the car comes instantaneously to rest when the spring is compressed by 1.4 m . The surface of the slope (including the section under the buffer) is now covered with gravel which exerts a constant resistive force of 7560 N on the car. The car is moving down the slope, and has speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is 24 m from the buffer, as shown in Fig. 2.3. It comes to rest when the spring has been compressed by \(x\) metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3ec81c4e-e0fa-43d9-9c79-ef9df746be8f-3_305_1087_2122_529} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
3. By considering work and energy, find the value of \(x\).

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).

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 \(\lambda\).

6. Two points \(A\) and \(B\) are in a vertical line, with \(A\) above \(B\) and \(A B = 4 a\). One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). Another light elastic spring, of natural length \(a\) and modulus of elasticity \(m g\), has one end attached to \(B\) and the other end attached to \(P\). The particle \(P\) hangs at rest in equilibrium.

1. Show that \(A P = \frac { 7 a } { 4 }\) The particle \(P\) is now pulled down vertically from its equilibrium position towards \(B\) and at time \(t = 0\) it is released from rest. At time \(t\), the particle \(P\) is moving with speed \(v\) and has displacement \(x\) from its equilibrium position. The particle \(P\) is subject to air resistance of magnitude \(m k v\), where \(k\) is a positive constant.
2. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \frac { 4 g } { a } x = 0$$
3. Find the range of values of \(k\) which would result in the motion of \(P\) being a damped oscillation.

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}

2 A light elastic string AB has stiffness \(k\). The end A is attached to a fixed point and a particle of mass \(m\) is attached at the end B . With the string vertical, the particle is released from rest from a point at a distance \(a\) below its equilibrium position. At time \(t\), the displacement of the particle below the equilibrium position is \(x\) and the velocity of the particle is \(v\).

1. Show that $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - k x$$
2. Show that, while the particle is moving upwards and the string is taut, $$v = - \sqrt { \frac { k } { m } \left( a ^ { 2 } - x ^ { 2 } \right) }$$
3. Hence use integration to find an expression for \(x\) at time \(t\) while the particle is moving upwards and the string is taut.

3 A uniform rigid rod AB of length \(2 a\) and mass \(m\) is smoothly hinged to a fixed point at A so that it can rotate freely in a vertical plane. A light elastic string of modulus \(\lambda\) and natural length \(a\) connects the midpoint of AB to a fixed point C which is vertically above A with \(\mathrm { AC } = a\). The rod makes an angle \(2 \theta\) with the upward vertical, where \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \pi\). This is shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system relative to A in terms of \(m , \lambda , a\) and \(\theta\). Show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 a \cos \theta ( 2 \lambda \sin \theta - 2 m g \sin \theta - \lambda ) .$$ Assume now that the system is set up so that the result (*) continues to hold when \(\pi < 2 \theta \leqslant \frac { 5 } { 3 } \pi\).
2. In the case \(\lambda < 2 m g\), show that there is a stable position of equilibrium at \(\theta = \frac { 1 } { 2 } \pi\). Show that there are no other positions of equilibrium in this case.
3. In the case \(\lambda > 2 m g\), find the positions of equilibrium for \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \frac { 5 } { 3 } \pi\) and determine for each whether the equilibrium is stable or unstable, justifying your conclusions.