6.02i Conservation of energy: mechanical energy principle

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.

8. A pendulum consists of a uniform rod \(P Q\), of mass \(3 m\) and length \(2 a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.

1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 33 } { 4 } m a ^ { 2 }\). The pendulum is released from rest in the position where \(P Q\) makes an angle \(\alpha\) with the downward vertical. At time \(t , P Q\) makes an angle \(\theta\) with the downward vertical.
2. Show that the angular speed, \(\dot { \theta }\), of the pendulum satisfies $$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a } .$$
3. Hence, or otherwise, find the angular acceleration of the pendulum. Given that \(\alpha = \frac { \pi } { 20 }\) and that \(P Q\) has length \(\frac { 8 } { 33 } \mathrm {~m}\),
4. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest. \section*{Advanced Level} \section*{Monday 25 June 2012 - Afternoon} \section*{Materials required for examination
   Mathematical Formulae (Pink)} Items included with question papers
   Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

4. A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane. The rod hangs in equilibrium with \(B\) below \(A\). The rod is rotated through a small angle and released from rest at time \(t = 0\).

1. Show that the motion of the rod is approximately simple harmonic.
2. Using this approximation, find the time \(t\) when the rod is first vertical after being released.
   (Total 6 marks)

7. At time \(t = 0\), a small body is projected vertically upwards. While ascending it picks up small drops of moisture from the atmosphere. The drops of moisture are at rest before they are picked up. At time \(t\), the combined body \(P\) has mass \(m\) and speed \(v\).

1. Show that, while \(P\) is moving upwards, \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + v \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The initial mass of \(P\) is \(M\), and \(m = M \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
2. Show that, while \(P\) is moving upwards, \(\frac { \mathrm { d } } { \mathrm { d } t } \left( v \mathrm { e } ^ { k t } \right) = - g \mathrm { e } ^ { k t }\). Given that the initial projection speed of \(P\) is \(\frac { g } { 2 k }\),
3. find, in terms of \(M\), the mass of \(P\) when it reaches its highest point.
   (Total 15 marks)

7. A uniform plane circular disc, of mass \(m\) and radius \(a\), hangs in equilibrium from a point \(B\) on its circumference. The disc is free to rotate about a fixed smooth horizontal axis which is in the plane of the disc and tangential to the disc at \(B\). A particle \(P\), of mass \(m\), is moving horizontally with speed \(u\) in a direction which is perpendicular to the plane of the disc. At time \(t = 0 , P\) strikes the disc at its centre and adheres to the disc.

1. Show that the angular speed of the disc immediately after it has been struck by \(P\) is \(\frac { 4 u } { 9 a }\).
   (6) It is given that \(u ^ { 2 } = \frac { 1 } { 10 } a g\), and that air resistance is negligible.
2. Find the angle through which the disc turns before it first comes to instantaneous rest. The disc first returns to its initial position at time \(t = T\).
3. 1. Write down an equation of motion for the disc.
   2. Hence find \(T\) in terms of \(a , g\) and \(m\), using a suitable approximation which should be justified.

5. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The rod is hanging in equilibrium with \(B\) below \(A\) when it is hit by a particle of mass \(m\) moving horizontally with speed \(v\) in a vertical plane perpendicular to the axis. The particle strikes the rod at \(B\) and immediately adheres to it.

1. Show that the angular speed of the rod immediately after the impact is \(\frac { 3 v } { 8 a }\). Given that the rod rotates through \(120 ^ { \circ }\) before first coming to instantaneous rest,
2. find \(v\) in terms of \(a\) and \(g\).
3. find, in terms of \(m\) and \(g\), the magnitude of the vertical component of the force acting on the \(\operatorname { rod }\) at \(A\) immediately after the impact.
   (5)

3. A uniform lamina of mass \(m\) is in the shape of a rectangle \(P Q R S\), where \(P Q = 8 a\) and \(Q R = 6 a\).

1. Find the moment of inertia of the lamina about the edge \(P Q\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-3_336_772_528_642}
   \end{figure} The flap on a letterbox is modelled as such a lamina. The flap is free to rotate about an axis along its horizontal edge \(P Q\), as shown in Fig. 1. The flap is released from rest in a horizontal position. It then swings down into a vertical position.
2. Show that the angular speed of the flap as it reaches the vertical position is \(\sqrt { \left( \frac { g } { 2 a } \right) }\).
3. Find the magnitude of the vertical component of the resultant force of the axis \(P Q\) on the flap, as it reaches the vertical position.

4. A uniform circular disc, of mass \(m\) and radius \(r\), has a diameter \(A B\). The point \(C\) on \(A B\) is such that \(A C = \frac { 1 } { 2 } r\). The disc can rotate freely in a vertical plane about a horizontal axis through \(C\), perpendicular to the plane of the disc. The disc makes small oscillations in a vertical plane about the position of equilibrium in which \(B\) is below \(A\).

1. Show that the motion is approximately simple harmonic.
2. Show that the period of this approximate simple harmonic motion is \(\pi \sqrt { \left( \frac { 6 r } { g } \right) }\). The speed of \(B\) when it is vertically below \(A\) is \(\sqrt { \left( \frac { g r } { 54 } \right) }\). The disc comes to rest when \(C B\) makes an angle \(\alpha\) with the downward vertical.
3. Find an approximate value of \(\alpha\).
   (3)

7. \begin{figure}[h] \end{figure} A body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), with a uniform rod. The centres of the discs are fixed to the ends \(A\) and \(B\) of the rod, which has mass \(3 m\) and length 8a. The discs and the rod are coplanar, as shown in Fig. 2. The body is free to rotate in a vertical plane about a smooth fixed horizontal axis. The axis is perpendicular to the plane of the discs and passes through the point \(O\) of the rod, where \(A O = 3 a\).

1. Show that the moment of inertia of the body about the axis is \(54 m a ^ { 2 }\). The body is held at rest with \(A B\) horizontal and is then released. When the body has turned through an angle of \(30 ^ { \circ }\), the rod \(A B\) strikes a small fixed smooth peg \(P\) where \(O P = 3 a\). Given that the body rebounds from the peg with its angular speed halved by the impact,
2. show that the magnitude of the impulse exerted on the body by the peg at the impact is $$9 m \sqrt { \left( \frac { 5 g a } { 6 } \right) } .$$ END

5. A uniform square lamina \(A B C D\), of mass \(m\) and side \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about \(L\) is \(\frac { 8 m a ^ { 2 } } { 3 }\). Given that the lamina is released from rest when the line \(A C\) makes an angle of \(\frac { \pi } { 3 }\) with the downward vertical,

1. find the magnitude of the vertical component of the force acting on the lamina at \(A\) when the line \(A C\) is vertical. Given instead that the lamina now makes small oscillations about its position of stable equilibrium,
2. find the period of these oscillations.
   (5)
   (Total 12 marks)

7. A uniform lamina of mass \(m\) is in the shape of an equilateral triangle \(A B C\) of perpendicular height \(h\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) through \(A\) and perpendicular to the lamina.

1. Show, by integration, that the moment of inertia of the lamina about \(L\) is \(\frac { 5 m h ^ { 2 } } { 9 }\). The centre of mass of the lamina is \(G\). The lamina is in equilibrium, with \(G\) below \(A\), when it is given an angular speed \(\sqrt { \left( \frac { 6 g } { 5 h } \right) }\).
2. Find the angle between \(A G\) and the downward vertical, when the lamina first comes to rest.
3. Find the greatest magnitude of the angular acceleration during the motion.
   (Total 17 marks)

6. \begin{figure}[h] \end{figure} A lamina \(S\) is formed from a uniform disc, centre \(O\) and radius \(2 a\), by removing the disc of centre \(O\) and radius \(a\), as shown in Figure 2. The mass of \(S\) is \(M\).

1. Show that the moment of inertia of \(S\) about an axis through \(O\) and perpendicular to its plane is \(\frac { 5 } { 2 } M a ^ { 2 }\).
   (3) The lamina is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) lies in the plane of \(S\) and is a tangent to its outer circumference, as shown in Figure 2.
2. Show that the moment of inertia of \(S\) about \(L\) is \(\frac { 21 } { 4 } M a ^ { 2 }\).
   (4) \(S\) is displaced through a small angle from its position of stable equilibrium and, at time \(t = 0\), it is released from rest. Using the equation of motion of \(S\), with a suitable approximation,
3. find the time when \(S\) first passes through its position of stable equilibrium.
   (6)

8. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis through the point \(X\) on the rod, where \(A X = \frac { 1 } { 2 } a\). A particle of mass \(m\) is attached to the rod at \(B\). At time \(t = 0\), the rod is vertical, with \(B\) above \(A\), and is given an initial angular speed \(\sqrt { \frac { g } { a } }\). When the rod makes an angle \(\theta\) with the upward vertical, the angular speed of the rod is \(\omega\), as shown in Figure 3.

1. By using the principle of the conservation of energy, show that $$\omega ^ { 2 } = \frac { g } { 2 a } ( 5 - 3 \cos \theta )$$
2. Find the angular acceleration of the rod when it makes an angle \(\theta\) with the upward vertical. When \(\theta = \phi\), the resultant force of the axis on the rod is in a direction perpendicular to the rod.
3. Find \(\cos \phi\).

6. A uniform circular disc, of radius \(r\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which is perpendicular to the plane of the disc and passes through a point which is \(\frac { 1 } { 4 } r\) from the centre of the disc. The disc is held at rest with its centre vertically above the axis. The disc is then slightly disturbed from its rest position. You may assume without proof that the moment of inertia of the disc about \(L\) is \(\frac { 9 m r ^ { 2 } } { 16 }\).

1. Show that the angular speed of the disc when it has turned through \(\frac { \pi } { 2 }\) is \(\sqrt { } \left( \frac { 8 g } { 9 r } \right)\).
2. Find the magnitude of the force exerted on the disc by the axis when the disc has turned through \(\frac { \pi } { 2 }\).

2. A uniform equilateral triangular lamina \(A B C\) has mass \(m\) and sides of length \(\sqrt { } 3 a\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\), which passes through \(A\) and is perpendicular to the lamina. The midpoint of \(B C\) is \(D\). The lamina is held with \(A D\) making an angle of \(60 ^ { \circ }\) with the upward vertical through \(A\) and released from rest. The moment of inertia of the lamina about the axis \(L\) is \(\frac { 5 m a ^ { 2 } } { 4 }\) Find the speed of \(D\) when \(A D\) is vertical.
(8)

A bead of mass 0.5 kg is threaded on a smooth straight wire. The forces acting on the bead are a constant force \(( 2 \mathbf { i } + 3 \mathbf { j } + \chi \mathbf { k } ) \mathrm { N }\), its weight \(( - 4.9 \mathbf { k } ) \mathrm { N }\), and the reaction on the bead from the wire.

1. Explain why the reaction on the bead from the wire does no work as the bead moves along the wire. The bead moves from the point \(A\) with position vector \(( \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\) relative to a fixed origin \(O\) to the point \(B\) with position vector \(( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\). The speed of the bead at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of the bead at \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the value of \(x\).
   

4 The diagram shows two points A and B on a snowy slope. A is a vertical distance of 25 m above B. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-5_220_1376_306_244} A rider and snowmobile, with a combined mass of 240 kg , start at the top of the slope, heading in the direction of \(B\). As the snowmobile passes \(A\), with a speed of \(3 \mathrm {~ms} ^ { - 1 }\), the rider switches off the engine so that the snowmobile coasts freely. When the snowmobile passes B, it has a speed of \(18 \mathrm {~ms} ^ { - 1 }\). The resistances to motion can be modelled as a single, constant force of magnitude 120 N .

1. Calculate the distance the snowmobile travels from A to B. The rider now turns the snowmobile around and brings it back to B, so that it faces up the slope. Starting from rest, the snowmobile ascends the slope so that it passes A with a speed of \(7 \mathrm {~ms} ^ { - 1 }\). It takes 30 seconds for the snowmobile to travel from B to A. The resistances to motion can still be modelled as a single, constant force of magnitude 120 N .
2. Show that the snowmobile develops an average power of 2856 W during this time. The snowmobile can develop a maximum power of 6000 W . At a later point in the journey, the rider and snowmobile reach a different slope inclined at \(12 ^ { \circ }\) to the horizontal. The resistances to motion can still be modelled as a single, constant force of magnitude 120 N .
3. Determine the maximum speed with which the rider and snowmobile can ascend. The power developed by a vehicle is sometimes given in the non-SI unit mechanical horsepower \(( \mathrm { hp } ) .1 \mathrm { hp }\) is the power required to lift 550 pounds against gravity, starting and ending at rest, by 1 foot in 1 second.
4. Given that 1 metre \(\approx 3.28\) feet and \(1 \mathrm {~kg} \approx 2.2\) pounds, determine the number of watts that are equivalent to 1 hp .

2 A ball P of mass \(m \mathrm {~kg}\) is held at a height of 12.8 m above a horizontal floor. P is released from rest and rebounds from the floor. After the first bounce, P reaches a maximum height of 5 m above the floor. Two models, A and B , are suggested for the motion of P .
Model A assumes that air resistance may be neglected.

1. Determine, according to model A , the coefficient of restitution between P and the floor. Model B assumes that the collision between P and the floor is perfectly elastic, but that work is done against air resistance at a constant rate of \(E\) joules per metre.
2. Show that, according to model \(\mathrm { B } , \mathrm { E } = \frac { 39 } { 89 } \mathrm { mg }\).
3. Show that both models predict that P will attain the same maximum height after the second bounce.

4 The diagram shows three beads, A, B and C, of masses \(0.3 \mathrm {~kg} , 0.5 \mathrm {~kg}\) and 0.7 kg respectively, threaded onto a smooth wire circuit consisting of two straight and two semi-circular sections. The circuit occupies a vertical plane, with the two straight sections horizontal and the upper section 0.45 m directly above the lower section. \includegraphics[max width=\textwidth, alt={}, center]{a87d62b8-406d-44cd-9ffa-384005329566-5_361_961_450_248} Initially, the beads are at rest. A and B are each given an impulse so that they move towards each other, A with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B with a speed of \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the subsequent collision between A and \(\mathrm { B } , \mathrm { A }\) is brought to rest.

1. Show that the coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Bead B next collides with C.
2. Show that the speed of B before this collision is \(4.37 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures. In this collision between B and C , B is brought to rest.
3. Determine whether C next collides with A or with B .
4. Explain why, if B has a greater mass than C , B could not be brought to rest in their collision.

5 In the diagram below, points \(\mathrm { A } , \mathrm { B }\) and C lie in the same vertical plane. The slope AB is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(\mathrm { AB } = 5 \mathrm {~m}\). The point B is a vertical distance of 6.5 m above horizontal ground. The point C lies on the horizontal ground. \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-6_601_1285_395_244} Starting at A , a particle P , of mass \(m \mathrm {~kg}\), moves along the slope towards B , under the action of a constant force \(\mathbf { F }\). The force \(\mathbf { F }\) has a magnitude of 50 N and acts at an angle of \(\theta ^ { \circ }\) to AB in the same vertical plane as A and B . When P reaches \(\mathrm { B } , \mathbf { F }\) is removed, and P moves under gravity landing at C . It is given that

• the speed of P at A is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• the speed of P at B is \(6 \mathrm {~ms} ^ { - 1 }\),
• the speed of P at C is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• 58 J of work is done against non-gravitational resistances as P moves from A to B ,
• 42 J of work is done against non-gravitational resistances as P moves from B to C .
  1. By considering the motion from B to C, show that \(m = 4.33\) correct to 3 significant figures.
  2. By considering the motion from A to B , determine the value of \(\theta\).
  3. Calculate the power of \(\mathbf { F }\) at the instant that P reaches B .

7 Fig. 7.1 shows one end of a light inextensible string attached to a block A of mass 4.4 kg . The other end of the string is attached to a block B of mass 5.2 kg . Block A is in contact with a smooth horizontal plane. The string is taut and passes over a small smooth pulley at the end of the plane. Block B is inside a hollow vertical tube and the vertical sides of B are in contact with the tube. Initially B is 1.6 m above the horizontal base of the tube. \begin{figure}[h] \end{figure} The blocks are released from rest. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut. Block B reaches the base of the tube with speed \(3.5 \mathrm {~ms} ^ { - 1 }\).

1. Given that the frictional force exerted by the tube on B is constant, use an energy method to show that the magnitude of this force is 14.21 N . Blocks A and B remain attached to the opposite ends of a light inextensible string, but A is now in contact with a rough plane inclined at \(\theta ^ { \circ }\) to the horizontal, as shown in Fig. 7.2. The string connecting A and B is taut and passes over a small smooth pulley at the top of the plane. Block B is inside the same hollow vertical tube as before with the vertical sides of B in contact with the tube. It may be assumed that the frictional force exerted by the tube on B remains unchanged. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-8_623_723_552_260} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure} The coefficient of friction between block A and the plane is \(\frac { 3 } { 11 }\).
   The blocks are released from rest, with block B 1.6 m above the base of the tube. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut.
2. Given that block B reaches the base of the tube with speed \(0.7 \mathrm {~ms} ^ { - 1 }\), show that \(\theta\) satisfies the equation \(3 \cos \theta + 11 \sin \theta = k\),
   where \(k\) is a constant to be determined. \section*{END OF QUESTION PAPER} \section*{}

6 A sack of beans of mass 40 kg is pulled from rest at point A up a non-uniform slope onto and along a horizontal platform. Fig. 6 shows this slope AB and the platform BC , which is a vertical distance of 12 m above A. \begin{figure}[h] \end{figure}

1. Calculate the gain in the gravitational potential energy of the sack when it is moved from A to the platform. The sack has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) by the time it reaches C at the far end of the platform. The total work done against friction in moving the sack from A to C is 484 J . There are no other resistances to the sack's motion.
2. Calculate the total work done in moving the sack between the points A and C . At point C , travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the sack starts to slide down a straight chute inclined at \(\alpha\) to the horizontal. Point D at the bottom of the chute is at the same vertical height as A , as shown in Fig. 6. The chute is rough and the coefficient of friction between the chute and the sack is 0.6 . During this part of the motion, again the only resistance to the motion of the sack is friction.
3. Use an energy method to calculate the value of \(\alpha\) given that the sack is travelling at \(3 \mathrm {~ms} ^ { - 1 }\) when it reaches D . For safety reasons the sack needs to arrive at D with a speed of less than \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The value of \(\alpha\) can be adjusted to try to achieve this.
4. (A) Find the range of values of \(\alpha\) which achieve a safe speed at D .
   (B) Comment on whether adjusting \(\alpha\) is a practical way of achieving a safe speed at D .

6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H . \begin{figure}[h] \end{figure}

1. Show that the particle passes through H without leaving the surface of the hemisphere. After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
2. State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q . The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
3. Find the value of \(\cos \theta\) correct to 3 significant figures.
4. Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.

3 A young woman wishes to make a bungee jump. One end of an elastic rope is attached to her safety harness. The other end is attached to the bridge from which she will jump. She calculates that the stretched length of the rope at the bottom of her motion should be 20 m , she knows that her weight is 576 N and the stiffness of the elastic rope is \(90 \mathrm { Nm } ^ { - 1 }\). She has to calculate the unstretched length of rope required to perform the jump safely. She models the situation by assuming the following.

• The rope is of negligible mass.
• Air resistance may be neglected.
• She is a particle.
• She moves vertically downwards from rest.
• Her starting point is level with the fixed end of the rope.
• The length she calculates for the rope does not include any extra for attaching the ends.
  1. (A) Show that the greatest extension of the rope, \(X\), satisfies the equation \(X ^ { 2 } = 256\).
     (B) Hence determine the natural length of rope she needs.
  2. To remain safe she wishes to be sure that, if air resistance is taken into account, the stretched length of the rope of natural length determined in part (i) will not be more than 20 m . Advise her on this point.

3 A rough circular hoop, with centre O and radius 1 m , is fixed in a vertical plane. A , B and C are points on the hoop such that A and C are at the same horizontal level as O , and OB makes an angle of \(25 ^ { \circ }\) above the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-4_650_729_404_251} A bead P of mass 0.3 kg is threaded onto the hoop. P is projected vertically downwards from A on two separate occasions.

• The first time, when P is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at B .
• The second time, when P is projected with a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at C .

The situation is modelled by assuming that during the motion of P the magnitude of the frictional force exerted by the hoop on P is constant.

1. Determine the value of \(v\).
2. Comment on the validity of the modelling assumption used in this question.