6.02j Conservation with elastics: springs and strings

7. \begin{figure}[h] \end{figure} Figure 4 shows a particle \(P\) projected from the point \(A\) up the line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 4 } { 5 } . P\) is projected with speed \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction between \(P\) and the plane is \(\frac { 4 } { 7 }\). Given that \(P\) first comes to rest at point \(B\),

1. use the Work-Energy principle to show that the distance \(A B\) is 1.4 m . The particle then slides back down the plane.
2. Find, correct to 2 significant figures, the speed of \(P\) when it returns to \(A\).

7 A particle of mass 0.8 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to a fixed point \(O\). The particle is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\), the particle is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy show that \(v ^ { 2 } = 39.2 + 19.6 x - 12.5 x ^ { 2 }\).
2. Hence find
   1. the maximum extension of the string,
   2. the maximum speed of the particle,
   3. the maximum magnitude of the acceleration of the particle.

5 A light elastic string of natural length 1.6 m has modulus of elasticity 120 N . One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle \(P\) of weight 1.5 N . The particle is released from rest at the point \(A\), which is 2.1 m vertically below \(O\). It comes instantaneously to rest at \(B\), which is vertically above \(O\).

1. Verify that the distance \(A B\) is 4 m .
2. Find the maximum speed of \(P\) during its upward motion from \(A\) to \(B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_351_442_303_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_394_648_260_1018} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string of length \(0.8 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.4 kg and 0.58 kg respectively, attached to its ends. The string passes over a smooth horizontal cylinder of radius 0.8 m , which is fixed with its axis horizontal and passing through a fixed point \(O\). The string is held at rest in a vertical plane perpendicular to the axis of the cylinder, with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the cylinder through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).

4 A smooth cylinder of radius \(a \mathrm {~m}\) is fixed with its axis horizontal and \(O\) is the centre of a cross-section. Particle \(P\), of mass 0.4 kg , and particle \(Q\), of mass 0.6 kg , are connected by a light inextensible string of length \(\pi a \mathrm {~m}\). The string is held at rest with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the crosssection through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 3.92 a ( 3 \theta - 2 \sin \theta )\) and find an expression in terms of \(\theta\) for the normal force of the cylinder on \(P\) at this time.
2. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(\sin \alpha = k \alpha\) where \(k\) is a constant to be found.

5 A particle \(P\), of mass 2.5 kg , is in equilibrium suspended from a fixed point \(A\) by a light elastic string of natural length 3 m and modulus of elasticity 36.75 N . Another particle \(Q\), of mass 1 kg , is released from rest at \(A\) and falls freely until it reaches \(P\) and becomes attached to it.

1. Show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(2 \sqrt { 2 } \mathrm {~ms} ^ { - 1 }\). The combined particles fall a further distance \(X \mathrm {~m}\) before coming to instantaneous rest.
2. Find a quadratic equation satisfied by \(X\), and show that it simplifies to \(35 X ^ { 2 } - 56 X - 80 = 0\).

6 A bungee jumper of mass 70 kg is joined to a fixed point \(O\) by a light elastic rope of natural length 30 m and modulus of elasticity 1470 N . The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

1. Find the distance fallen by the jumper when maximum speed is reached.
2. Show that this maximum speed is \(26.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. Find the extension of the rope when the jumper is at the lowest position. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_543_616_310_301} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_668_709_267_1135} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A smooth horizontal cylinder of radius 0.6 m is fixed with its axis horizontal and passing through a fixed point \(O\). A light inextensible string of length \(0.6 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.3 kg and 0.4 kg respectively, attached at its ends. The string passes over the cylinder and is held at rest with \(P , O\) and \(Q\) in a straight horizontal line (see Fig. 1). The string is released and \(Q\) begins to descend. When the line \(O P\) makes an angle \(\theta\) radians, \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), with the horizontal, the particles have speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).
4. By considering the total energy of the system, or otherwise, show that $$v ^ { 2 } = 6.72 \theta - 5.04 \sin \theta .$$
5. Show that the magnitude of the contact force between \(P\) and the cylinder is $$( 5.46 \sin \theta - 3.36 \theta ) \text { newtons. }$$ Hence find the value of \(\theta\) for which the magnitude of the contact force is greatest.
6. Find the transverse component of the acceleration of \(P\) in terms of \(\theta\).

2

1. A fixed solid sphere with a smooth surface has centre O and radius 0.8 m . A particle P is given a horizontal velocity of \(1.2 \mathrm {~ms} ^ { - 1 }\) at the highest point on the sphere, and it moves on the surface of the sphere in part of a vertical circle of radius 0.8 m .
   1. Find the radial and tangential components of the acceleration of P at the instant when OP makes an angle \(\frac { 1 } { 6 } \pi\) radians with the upward vertical. (You may assume that P is still in contact with the sphere.)
   2. Find the speed of P at the instant when it leaves the surface of the sphere.
2. Two fixed points R and S are 2.5 m apart with S vertically below R . A particle Q of mass 0.9 kg is connected to R and to S by two light inextensible strings; Q is moving in a horizontal circle at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with both strings taut. The radius of the circle is 2.4 m and the centre C of the circle is 0.7 m vertically below S, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-2_547_720_1946_644} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Find the tension in the string RQ and the tension in the string \(S Q\).

7 \includegraphics[max width=\textwidth, alt={}, center]{a9e010ce-c3a8-4f95-a154-fd16ef3e5e5b-4_622_767_269_689} Particles \(P\) and \(Q\), with masses \(3 m\) and \(2 m\) respectively, are connected by a light inextensible string passing over a smooth light pulley. The particle \(P\) is connected to the floor by a light spring \(S _ { 1 }\) with natural length \(a\) and modulus of elasticity mg . The particle \(Q\) is connected to the floor by a light spring \(S _ { 2 }\) with natural length \(a\) and modulus of elasticity \(2 m g\). The sections of the string not in contact with the pulley, and the two springs, are vertical. Air resistance may be neglected. The particles \(P\) and \(Q\) move vertically and the string remains taut; when the length of \(S _ { 1 }\) is \(x\), the length of \(S _ { 2 }\) is ( \(3 a - x\) ) (see diagram).

1. Find the total potential energy of the system (taking the floor as the reference level for gravitational potential energy). Hence show that \(x = \frac { 4 } { 3 } a\) is a position of stable equilibrium.
2. By differentiating the energy equation, and substituting \(x = \frac { 4 } { 3 } a + y\), show that the motion is simple harmonic, and find the period.

1 A light elastic string has one end fixed to a vertical pole at A . The string passes round a smooth horizontal peg, P , at a distance \(a\) from the pole and has a smooth ring of mass \(m\) attached at its other end B . The ring is threaded onto the pole below A . The ring is at a distance \(y\) below the horizontal level of the peg. This situation is shown in Fig. 1. \begin{figure}[h] \end{figure} The string has stiffness \(k\) and natural length equal to the distance AP .

1. Express the extension of the string in terms of \(y\) and \(a\). Hence find the potential energy of the system relative to the level of P .
2. Use the potential energy to find the equilibrium position of the system, and show that it is stable.
3. Calculate the normal reaction exerted by the pole on the ring in the equilibrium position.

4 A uniform smooth pulley can rotate freely about its axis, which is fixed and horizontal. A light elastic string AB is attached to the pulley at the end B . The end A is attached to a fixed point such that the string is vertical and is initially at its natural length with B at the same horizontal level as the axis. In this position a particle P is attached to the highest point of the pulley. This initial position is shown in Fig. 4.1. The radius of the pulley is \(a\), the mass of P is \(m\) and the stiffness of the string AB is \(\frac { m g } { 10 a }\). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Fig. 4.2 shows the system with the pulley rotated through an angle \(\theta\) and the string stretched. Write down the extension of the string and hence find the potential energy, \(V\), of the system in this position. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \left( \frac { 1 } { 10 } \theta - \sin \theta \right)\).
2. Hence deduce that the system has a position of unstable equilibrium at \(\theta = 0\).
3. Explain how your expression for \(V\) relies on smooth contact between the string and the pulley. Fig. 4.3 shows the graph of the function \(\mathrm { f } ( \theta ) = \frac { 1 } { 10 } \theta - \sin \theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{636e1d23-3bbb-469f-8fc9-1f64da865126-3_538_1342_1706_404} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
4. Use the graph to give rough estimates of three further values of \(\theta\) (other than \(\theta = 0\) ) which give positions of equilibrium. In each case, state with reasons whether the equilibrium is stable or unstable.
5. Show on a sketch the physical situation corresponding to the least value of \(\theta\) you identified in part (iv). On your sketch, mark clearly the positions of P and B .
6. The equation \(\mathrm { f } ( \theta ) = 0\) has another root at \(\theta \approx - 2.9\). Explain, with justification, whether this necessarily gives a position of equilibrium.

2 A uniform rigid rod AB of mass \(m\) and length \(4 a\) is freely hinged at the end A to a horizontal rail. The end B is attached to a light elastic string BC of modulus \(\frac { 1 } { 2 } m g\) and natural length \(a\). The end C of the string is attached to a ring which is small, light and smooth. The ring can slide along the rail and is always vertically above B . The angle that AB makes below the rail is \(\theta\). The system is shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system when the string is stretched and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 4 m g a \cos \theta ( 2 \sin \theta - 1 )$$
2. Hence find any positions of equilibrium of the system and investigate their stability.

3 A uniform rod AB of mass \(m\) and length \(4 a\) is hinged at a fixed point C , where \(\mathrm { AC } = a\), and can rotate freely in a vertical plane. A light elastic string of natural length \(2 a\) and modulus \(\lambda\) is attached at one end to B and at the other end to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance \(2 a\) above C . The string is always vertical. This system is shown in Fig. 3 with the rod inclined at \(\theta\) to the horizontal. \begin{figure}[h] \end{figure}

1. Find an expression for \(V\), the potential energy of the system relative to C , and show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( \frac { 9 } { 2 } \lambda \sin \theta - m g \right)\).
2. Determine the positions of equilibrium and the nature of their stability in the cases
   (A) \(\lambda > \frac { 2 } { 9 } m g\),
   (B) \(\lambda < \frac { 2 } { 9 } m g\),
   (C) \(\lambda = \frac { 2 } { 9 } m g\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_522_755_342_696} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
3. Show, by integration, that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M a ^ { 2 }\). [You may assume the standard formula for the moment of inertia of a uniform circular disc about its axis of symmetry and the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] A frustum is made by taking a uniform cone of base radius 0.1 m and height 0.2 m and removing a cone of height 0.1 m and base radius 0.05 m as shown in Fig. 4.2. The mass of the frustum is 2.8 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_391_517_1352_813} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} The frustum can rotate freely about its axis of symmetry which is fixed and vertical.
4. Show that the moment of inertia of the frustum about its axis of symmetry is \(0.0093 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The frustum is accelerated from rest for \(t\) seconds by a couple of magnitude 0.05 N m about its axis of symmetry, until it is rotating at \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
5. Calculate \(t\).
6. Find the position of G , the centre of mass of the frustum. The frustum (rotating at \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) ) then receives an impulse tangential to the circumference of the larger circular face. This reduces its angular speed to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
7. To reduce its angular speed further, a parallel impulse of the same magnitude is now applied tangentially in the horizontal plane through G at the curved surface of the frustum. Calculate the resulting angular speed.

2 A small ring of mass \(m\) can slide freely along a fixed smooth horizontal rod. A light elastic string of natural length \(a\) and stiffness \(k\) has one end attached to a point A on the rod and the other end attached to the ring. An identical elastic string has one end attached to the ring and the other end attached to a point B which is a distance \(a\) vertically above the rod and a horizontal distance \(2 a\) from the point A . The displacement of the ring from the vertical line through B is \(x\), as shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Find an expression for \(V\), the potential energy of the system when \(0 < x < a\), and show that $$\frac { \mathrm { d } V } { \mathrm {~d} x } = 2 k x - k a - \frac { k a x } { \sqrt { a ^ { 2 } + x ^ { 2 } } }$$
2. Show that \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) is always positive.
3. Show that there is a position of equilibrium with \(\frac { 1 } { 2 } a < x < a\). State, with a reason, whether it is stable or unstable.

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.

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 .

4 A block of mass 20 kg is placed on a rough plane inclined at an angle \(30 ^ { \circ }\) to the horizontal. The block is pulled up the plane by a constant force acting parallel to a line of greatest slope.
The block passes through points A and B on the plane with speeds \(9 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively with B higher up the plane than A . The distance between A and B is \(x \mathrm {~m}\) and the coefficient of friction between the block and the plane is \(\frac { \sqrt { 3 } } { 49 }\). Use an energy method to determine the range of possible values of \(x\).

1. A particle \(P\), of mass \(m\), is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity kmg.

The other end of the spring is attached to a fixed point \(O\) on a ceiling.
The point \(A\) is vertically below \(O\) such that \(O A = 3 a\) The point \(B\) is vertically below \(O\) such that \(O B = \frac { 1 } { 2 } a\) The particle is held at rest at \(A\), then released and first comes to instantaneous rest at the point \(B\).

1. Show that \(k = \frac { 4 } { 3 }\)
2. Find, in terms of \(g\), the acceleration of \(P\) immediately after it is released from rest at \(A\).
3. Find, in terms of \(g\) and \(a\), the maximum speed attained by \(P\) as it moves from \(A\) to \(B\).

1. A light elastic string with natural length \(l\) and modulus of elasticity \(k m g\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\).

The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The package is initially held at \(A\). The package is then projected with speed \(\sqrt { 6 g l }\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(A B = 31\).
The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\) By modelling the package as a particle,

1. show that \(k = \frac { 15 } { 26 }\)
2. find the acceleration of the package at the instant it starts to move back down the plane from the point \(B\).

6. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(3 l\) and modulus of elasticity \(3 m g\).
One end of the spring is attached to a fixed point \(X\) on a rough inclined plane.
The other end of the spring is attached to a package \(P\) of mass \(m\).
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) The package is initially held at the point \(Y\) on the plane, where \(X Y = l\). The point \(Y\) is higher than \(X\) and \(X Y\) is a line of greatest slope of the plane, as shown in Figure 2. The package is released from rest at \(Y\) and moves up the plane.
The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 }\) By modelling \(P\) as a particle,

1. show that the acceleration of \(P\) at the instant when \(P\) is released from rest is \(\frac { 17 } { 15 } \mathrm {~g}\)
2. find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the spring first reaches its natural length of 31 .

6. \begin{figure}[h] \end{figure} Two blocks, \(A\) and \(B\), of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string. Initially \(A\) is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\) The string passes over a small smooth light pulley \(P\) that is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(A\) is held on the plane with the distance \(A P\) greater than 3 m .
Block \(B\) hangs freely below \(P\) at a distance of 3 m above the ground, as shown in Figure 4. The coefficient of friction between \(A\) and the plane is \(\mu\) Block \(A\) is released from rest with the string taut.
By modelling the blocks as particles,

1. find the potential energy lost by the whole system as a result of \(B\) falling 3 m . Given that the speed of \(B\) at the instant it hits the ground is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and ignoring air resistance,
2. use the work-energy principle to find the value of \(\mu\) After \(B\) hits the ground, \(A\) continues to move up the plane but does not reach the pulley in the subsequent motion.
   Block \(A\) comes to instantaneous rest after moving a total distance of ( \(3 + d\) ) m from its point of release. Ignoring air resistance,
3. use the work-energy principle to find the value of \(d\) \includegraphics[max width=\textwidth, alt={}, center]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_2255_50_309_1981}

8 A bungee jumper, of mass 49 kg , is attached to one end of a light elastic cord of natural length 22 metres and modulus of elasticity 1078 newtons. The other end of the cord is attached to a horizontal platform, which is at a height of 60 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached, and falls vertically. The bungee jumper can be modelled as a particle. Assume that Hooke's Law applies whilst the cord is taut and that air resistance is negligible throughout the motion. When the bungee jumper has fallen \(x\) metres, his speed is \(v \mathrm {~ms} ^ { - 1 }\).

1. By considering energy, show that, when \(x\) is greater than 22, $$5 v ^ { 2 } = 318 x - 5 x ^ { 2 } - 2420$$
2. Explain why \(x\) must be greater than 22 for the equation in part (a) to be valid. ( 1 mark)
3. Find the maximum value of \(x\).
4. 1. Show that the speed of the bungee jumper is a maximum when \(x = 31.8\).
   2. Hence find the maximum speed of the bungee jumper.

8

1. Hooke's law states that the tension in a stretched string of natural length \(l\) and modulus of elasticity \(\lambda\) is \(\frac { \lambda x } { l }\) when its extension is \(x\). Using this formula, prove that the work done in stretching a string from an unstretched position to a position in which its extension is \(e\) is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
   (3 marks)
2. A particle, of mass 5 kg , is attached to one end of a light elastic string of natural length 0.6 metres and modulus of elasticity 150 N . The other end of the string is fixed to a point \(O\).
   1. Find the extension of the elastic string when the particle hangs in equilibrium directly below \(O\).
   2. The particle is pulled down and held at the point \(P\), which is 0.9 metres vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 11.25 J .
   3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres above \(\boldsymbol { P }\). Show that, while the string is taut, $$v ^ { 2 } = 10.4 x - 50 x ^ { 2 }$$
   4. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.

6 A block, of mass 5 kg , is attached to one end of a length of elastic string. The other end of the string is fixed to a vertical wall. The block is placed on a horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 180 N . The block is pulled so that it is 2 m from the wall and is then released from rest. Whilst taut, the string remains horizontal. It may be assumed that, after the string becomes slack, it does not interfere with the movement of the block. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-5_396_960_660_534}

1. Calculate the elastic potential energy when the block is 2 m from the wall.
2. If the horizontal surface is smooth, find the speed of the block when it hits the wall.
3. The surface is in fact rough and the coefficient of friction between the block and the surface is \(\mu\). Find \(\mu\) if the block comes to rest just as it reaches the wall.

8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A 'reverse' bungee jump consists of two identical elastic ropes. One end of each elastic rope is attached to either side of the top of a gorge. The other ends are both attached to Hannah, who has mass 84 kg
Hannah is modelled as a particle, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-12_467_844_678_598} The depth of the gorge is 50 metres and the width of the gorge is 40 metres.
Each elastic rope has natural length 30 metres and modulus of elasticity 3150 N
Hannah is released from rest at the centre of the bottom of the gorge.
8

1. Show that the speed of Hannah when the ropes become slack is \(30 \mathrm {~ms} ^ { - 1 }\) correct to two significant figures.
   8
2. Determine whether Hannah is moving up or down when the ropes become taut again. [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-14_2492_1721_217_150} Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin.