6.02i Conservation of energy: mechanical energy principle

8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Omar, a bungee jumper of mass 70 kg , has his ankles attached to one end of an elastic cord. The other end of the cord is attached to a bridge which is 80 metres above the surface of a river. Omar steps off the bridge at the point where the cord is attached and falls vertically downwards. The cord can be modelled as a light elastic string of natural length \(L\) metres and modulus of elasticity 2800 N Model Omar as a particle. 8

1. Given that Omar just reaches the surface of the river before being pulled back up, find the value of \(L\) Fully justify your answer.
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2. If Omar is not modelled as a particle, explain the effect of revising this assumption on your answer to part (a).

13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart. \(A B C\) is a horizontal line. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13

1. Show that the particle moves with simple harmonic motion.
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2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
   [0pt] [4 marks]

4 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) An inelastic string has length 1.2 metres.
One end of the string is attached to a fixed point \(O\).
A sphere, of mass 500 grams, is attached to the other end of the string.
The sphere is held, with the string taut and at an angle of \(20 ^ { \circ }\) to the vertical, touching the chin of a student, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-04_739_511_676_762} The sphere is released from rest.
Assume that the student stays perfectly still once the sphere has been released.
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1. Calculate the maximum speed of the sphere.
   

   4

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.
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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.
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   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.
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7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A box, of mass 8 kg , is on a rough horizontal surface.
A string attached to the box is used to pull it along the surface.
The string is inclined at an angle of \(40 ^ { \circ }\) above the horizontal.
The tension in the string is 50 newtons.
As the box moves a distance of \(x\) metres, its speed increases from \(2 \mathrm {~ms} ^ { - 1 }\) to \(5 \mathrm {~ms} ^ { - 1 }\) The coefficient of friction between the box and the surface is 0.4
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1. By using an energy method, find \(x\).
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2. Describe how the model could be refined to obtain a more realistic value of \(x\) and use an energy argument to explain whether this would increase or decrease the value of \(x\).

5 A train of mass 10000 kg is travelling at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a buffer. The buffer brings the train to rest. As the buffer brings the train to rest it compresses by 0.2 metres.
When the buffer is compressed by a distance of \(x\) metres it exerts a force of magnitude \(F\) newtons, where $$F = A x + 9000 x ^ { 2 }$$ where \(A\) is a constant. 5

1. Find, in terms of \(A\), the work done in compressing the buffer by 0.2 metres.
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2. Find the value of \(A\)

8

1. Use an energy method to find the maximum speed of the crate.
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2. Use an energy method to find the total distance travelled by the crate.
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3. A student claims that in reality the crate is unlikely to travel more than 5.3 metres in total. Comment on the validity of this claim. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-12_2488_1732_219_139}

1 \includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-02_494_390_251_255} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).

2 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m . \begin{enumerate}[label=(\alph*)] \item Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\). \item Find the value of \(R\). \item Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns . \item

1. Find the speed of \(Q\) after the collision.
2. Hence show that the collision is inelastic. It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b9741472-f230-4e2d-9c8b-47f7e168e938-03_355_565_269_274} In a model, the following modelling assumptions are made.
   For a particular portion of banked track, \(r = 24\).
   (b) Find the value of \(v\) as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.
   (c) Explain

3 A right circular cone \(C\) of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of \(C\). The other end of the string is attached to a particle \(P\) of mass 2.5 kg . \(P\) moves in a horizontal circle with constant speed and in contact with the smooth curved surface of \(C\). The extension of the string is 1.5 m .

1. Find the tension in the string.
2. Find the speed of \(P\).

1 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.

2 One end of a light inextensible string of length 0.75 m is attached to a particle \(A\) of mass 2.8 kg . The other end of the string is attached to a fixed point \(O . A\) is projected horizontally with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 0.75 m vertically above \(O\) (see Fig. 2). When \(O A\) makes an angle \(\theta\) with the upward vertical the speed of \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \(\xrightarrow [ A \text { a } ] { 6 \mathrm {~m} \mathrm {~s} ^ { - 1 } }\) Fig. 2

1. Show that \(v ^ { 2 } = 50.7 - 14.7 \cos \theta\).
2. Given that the string breaks when the tension in it reaches 200 N , find the angle that \(O A\) turns through between the instant that \(A\) is projected and the instant that the string breaks.

2 One end of a light inextensible string of length 0.8 m is attached to a fixed point, \(O\). The other end is attached to a particle \(P\) of mass \(1.2 \mathrm {~kg} . P\) hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below \(O\) is \(F\). \(P\) is projected horizontally with speed \(3.5 \mathrm {~ms} ^ { - 1 }\). The string breaks when \(O P\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{0428f2f2-12c4-4e89-93ab-8cfe2c5aca4a-02_757_889_1482_251}

1. Find the magnitude of the tension in the string at the instant before the string breaks.
2. Find the distance between \(F\) and the point where \(P\) first hits the plane.

7 \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_339_511_349_817} A particle of mass 0.3 kg is attached to one end \(A\) of a light inextensible string of length 1.5 m . The other end \(B\) of the string is attached to a ceiling, so that the particle may swing in a vertical plane. The particle is released from rest when the string is taut and makes an angle of \(75 ^ { \circ }\) with the vertical (see diagram). Air resistance may be regarded as being negligible.

1. Show that, at an instant when the string makes an angle of \(40 ^ { \circ }\) with the vertical, the speed of the particle is \(3.90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
2. By considering Newton's second law, along and perpendicular to the string, find the radial and transverse components of acceleration, at this same instant, and hence the magnitude of the acceleration of the particle. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-4_419_604_1370_772} A smooth sphere of mass 0.3 kg is moving in a straight line on a horizontal surface. It collides with a vertical wall when the velocity of the sphere is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the wall (see diagram). The coefficient of restitution between the sphere and the wall is 0.4 .
3. (a) Find the component of the velocity of the sphere perpendicular to the wall immediately after the collision.
   (b) Find the magnitude of the impulse exerted by the wall on the sphere.
4. Determine the magnitude and direction of the velocity of the sphere immediately after the collision, giving the direction as an acute angle to the wall.

9 At noon a vessel, \(A\), leaves a port, \(O\), and travels at \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(042 ^ { \circ }\). Also at noon a second vessel, \(B\), leaves another port, \(P , 13 \mathrm {~km}\) due north of \(O\), and travels at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(090 ^ { \circ }\). Take \(O\) as the origin and \(\mathbf { i }\) and \(\mathbf { j }\) as unit vectors east and north respectively.

1. Express the velocity vector of \(A\) relative to \(B\) in the form \(a \mathbf { i } + b \mathbf { j }\), where \(a\) and \(b\) are constants to be determined.
2. Express the position vector of \(A\) relative to \(B\), at time \(t\) hours after the vessels have left port, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
3. Explain why the scalar product of the vectors in parts (i) and (ii) is zero when the two vessels are closest together.
4. Find the time at which the two vessels are closest together. \(10 A\) and \(B\) are two points 6 m apart on a smooth horizontal surface. A particle, \(P\), of mass 0.5 kg is attached to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 20 N , and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 10 N , such that \(P\) is between \(A\) and \(B\).

1. Find the length \(A P\) when \(P\) is in equilibrium. \(P\) is held at the point \(C\), where \(C\) is between \(A\) and \(B\) and \(A C = 4.5 \mathrm {~m} . P\) is then released from rest. At time \(t\) seconds after being released, the displacement of \(P\) from the equilibrium position is \(y\) metres.
2. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } = - 40 y$$
3. Find the time taken for \(P\) to reach the mid-point of \(A B\) for the first time. \includegraphics[max width=\textwidth, alt={}, center]{963c0834-fe49-480b-9bb5-1ace4254641a-6_750_1187_258_479} Two particles, \(P\) and \(Q\), are projected simultaneously from the same point on a plane inclined at \(\alpha\) to the horizontal. \(P\) is projected up the plane and \(Q\) down the plane. Each particle is projected with speed \(V\) at an angle \(\theta\) to the plane. Both particles move in a vertical plane containing a line of greatest slope of the inclined plane and you are given that \(\alpha + \theta < \frac { 1 } { 2 } \pi\) (see diagram).

1. Show that the range of \(P\), up the plane, is given by $$\frac { 2 V ^ { 2 } \sin \theta } { g \cos ^ { 2 } \alpha } ( \cos \theta \cos \alpha - \sin \theta \sin \alpha ) .$$
2. Write down a similar expression for the range of \(Q\), down the plane.
3. Given that the range up the plane is a quarter of the range down the plane and that \(\alpha = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\), find \(\theta\).

11 Two light strings, each of natural length \(a\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 a\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that $$\ddot { x } = - \frac { 12 g x } { a } \left( 1 - \frac { a } { \sqrt { 9 a ^ { 2 } + x ^ { 2 } } } \right) .$$
2. Given that \(\frac { x } { a }\) is small throughout the motion, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { a }$$ and state the period of the simple harmonic motion that this equation represents.
3. Given that the initial speed of \(P\) is \(\sqrt { \frac { g a } { 200 } }\), show that the amplitude of the simple harmonic motion is \(\frac { 1 } { 40 } a\).

11 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_384_524_587_808} One end of a light inextensible string of length 10 m is attached to a fixed point \(O\). A particle of mass 5 kg is attached to the other end of the string. The particle rests in equilibrium below \(O\). The particle is pulled aside until the string makes an angle of \(60 ^ { \circ }\) with the downward vertical and released from rest (see diagram). At the instant when the string makes an angle \(\cos ^ { - 1 } \left( \frac { 4 } { 5 } \right)\) with the downward vertical,

1. find the speed of the particle and the tension in the string,
2. show that the magnitude of the acceleration of the particle is \(6 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

9 A light string, of natural length 0.5 m and modulus of elasticity 4 N , has one end attached to the ceiling of a room. A particle of mass 0.2 kg is attached to the free end of the string and hangs in equilibrium.

1. Find the extension of the string when the particle is in the equilibrium position. The particle is pulled down a further 0.5 m from the equilibrium position and released from rest. At time \(t\) seconds the displacement of the particle from the equilibrium position is \(x \mathrm {~m}\).
2. Show that, while the string is taut, the equation of motion is \(\ddot { x } = - 40 x\).
3. Find the time taken for the string to become slack for the first time.
4. Show that the particle comes to instantaneous rest 0.125 m below the ceiling.

10 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the free end of the string and the particle hangs at rest vertically below \(O\). The particle is projected horizontally with speed \(u\).

1. Find the tension in the string when it makes an angle \(\theta\) with the downward vertical, whilst the string remains taut.
2. Deduce that the particle will perform complete circles provided that \(u ^ { 2 } \geqslant 5 a g\).
3. It is given that \(u ^ { 2 } = 4 a g\). Find
   1. the tension in the string when \(\theta = 60 ^ { \circ }\),
   2. the value of \(\theta\), to the nearest degree, at the instant when the string becomes slack.

8 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-4_182_803_264_671} A light spring of modulus of elasticity 8 N and natural length 0.4 m has one end fixed to a smooth horizontal table at a fixed point \(L\). A particle of mass 0.2 kg is attached to the other end of the spring and pulled out horizontally to a point \(M\) on the table, so that the spring is extended by 0.2 m . The particle is then released from rest. The mid-point of \(L M\) is \(N\) and the point \(O\) is on \(L M\) such that \(L O = 0.4 \mathrm {~m}\) (see diagram).

1. Show that the particle moves in simple harmonic motion with centre \(O\) and state the exact period of its motion.
2. Find the exact time taken for the particle to move directly from \(M\) to \(N\).

8 A rough plane is inclined at \(20 ^ { \circ }\) to the horizontal. A particle of mass 0.4 kg is projected down the plane, along a line of greatest slope, at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After it has travelled 3 m down the plane its speed is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By considering the change in energy, find the magnitude of the frictional force, assumed constant.

14 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 3 N is attached to a ceiling at a point \(P\). A particle of mass 0.3 kg is attached to the other end of the string.

1. Find the extension of the string when the particle hangs vertically in equilibrium. The particle is released from rest at \(P\) so that it falls vertically. Find
2. the maximum extension of the string,
3. the equation of motion for the particle when the string is stretched, in terms of the displacement \(x \mathrm {~m}\) below the equilibrium position,
4. the time between the string first becoming stretched and next becoming unstretched again.

7 A child of mass 20 kg slides down a rough slope of length 16 m against a constant frictional force \(F \mathrm {~N}\). Starting with an initial speed of \(2 \mathrm {~ms} ^ { - 1 }\) at a point 8 m above the ground, she reaches the ground with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(F\).

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right) .$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

13 A fairground game consists of a small ball fixed to one end of a light inextensible string of length 0.6 m . The other end of the string is fixed to a point \(O\). A small bell is fixed 0.6 m vertically above \(O\). Initially the ball hangs vertically in equilibrium. The object of the game is to project the ball with an initial horizontal velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it moves in a vertical circle and hits the bell.

1. Find the smallest possible value of \(u\) for which the ball hits the bell.
2. Given, instead, that the value of \(u\) is 5 , find the angle made by the string with the upward vertical at the moment when the string becomes slack.