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

4 \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

1 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A particle of mass 2 kg is attached to one end of a light elastic string of natural length 0.5 metres and modulus of elasticity 100 N . The other end of the string is attached to the point \(O\). Find the extension of the elastic string when the particle hangs in equilibrium vertically below \(O\). Circle your answer.
0.01 m
0.1 m
0.2 m
0.4 m

8 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point \(O\).
The particle is pulled down and released from rest at a point 0.6 metres directly below \(O\).
The particle then moves vertically and next comes to rest when it is 0.1 metres below \(O\).
Assume that no air resistance acts on the particle.
8

1. Find the maximum speed of the particle.
   [0pt] [6 marks]
   8
2. Describe one way in which the model you have used could be refined.

4 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 \(P\).

2 The coordinates of two points, \(A\) and \(B\), are \(( - 1,6 )\) and \(( 5,12 )\) respectively, where the units of the coordinate axes are metres. A particle \(P\) moves from \(A\) to \(B\) under the action of several forces. The force \(\mathbf { F } = 7 \mathbf { i } - 2 \mathbf { j } \mathbf { N }\) is one of the forces acting on \(P\).

1. Calculate the work done by \(\mathbf { F }\) on \(P\) as \(P\) moves from \(A\) to \(B\). At the instant when \(P\) reaches \(B\) its velocity is \(- \mathbf { i } - 5 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
2. Find the power generated by \(\mathbf { F }\) at the instant that \(P\) reaches \(B\). One end of a light elastic string was attached to the origin of the coordinate system and the other to \(P\) when \(P\) was at \(A\), before it moved to \(B\). The natural length of the string is 8 m and its modulus of elasticity is 24 N .
3. At the instant that \(P\) reaches \(B\), find the following.

2 One end of a light elastic string of natural length 1.4 m and modulus of elasticity 20 N is attached to a small object \(B\) of mass 2.5 kg . The other end of the string is attached to a fixed point \(O\). Object \(B\) is projected vertically upwards from \(O\) with a speed of \(u \mathrm {~ms} ^ { - 1 }\).

1. State one assumption required to model the motion of \(B\). The greatest height above \(O\) achieved by \(B\) is 8.1 m .
2. Determine the value of \(u\).

8 A shape, \(S\), is formed by attaching a particle of mass \(2 m \mathrm {~kg}\) to the vertex of a uniform solid cone of mass \(8 m \mathrm {~kg}\). The height of the cone is \(h \mathrm {~m}\) and the radius of the base of the cone is 1.1 m .

1. Explain why the centre of mass of \(S\) must lie on the central axis of the cone. Two strings are attached to \(S\), one at the vertex of the cone and one at \(A\) which is a point on the edge of the base of \(S\). The other ends of the strings are attached to a horizontal ceiling in such a way that the strings are both vertical. The string attached to \(S\) at \(A\) is inextensible and has length 1.6 m . The string attached to \(S\) at the vertex is elastic with modulus of elasticity 8 mgN . Shape \(S\) is in equilibrium with its axis horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-6_654_1541_879_244}
2. Determine the natural length of the elastic string.

2 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.

8 One end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\) is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\) which is on a horizontal surface. The surface is modelled as being smooth and \(A\) moves in a circular path around \(O\) with constant speed \(v \mathrm {~ms} ^ { - 1 }\). The extension of the string is denoted by \(x \mathrm {~m}\).

1. Show that \(x\) satisfies \(\lambda x ^ { 2 } + \lambda | x - | m v ^ { 2 } = 0\).
2. By solving the equation in part (a) and using a binomial series, show that if \(\lambda\) is very large then \(\lambda \mathrm { x } \approx \mathrm { mv } ^ { 2 }\).
3. By considering the tension in the string, explain how the result obtained when \(\lambda\) is very large relates to the situation when the string is inextensible. The nature of the horizontal surface is such that the modelling assumption that it is smooth is justifiable provided that the speed of the particle does not exceed \(7 \mathrm {~ms} ^ { - 1 }\). In the case where \(m = 0.16\) and \(\lambda = 260\), the extension of the string is measured as being 3.0 cm .
4. Estimate the value of \(v\).
5. Explain whether the value of \(v\) means that the modelling assumption is necessarily justifiable in this situation.

1 One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass 0.4 kg . \(O\) is a vertical distance of 1 m below a horizontal ceiling. \(P\) is held at a point 1.5 m vertically below \(O\) and released from rest (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-2_470_371_593_255} Assuming that there is no obstruction to the motion of \(P\) as it passes \(O\), find the speed of \(P\) when it first hits the ceiling.

2 As part of a training exercise an army recruit of mass 75 kg falls a vertical distance of 5 m before landing on a mat of thickness 1.2 m . The army recruit sinks a distance of \(x \mathrm {~m}\) into the mat before instantaneously coming to rest. The mat can be modelled as a spring of natural length 1.2 m and modulus of elasticity 10800 N and the army recruit can be modelled as a particle falling vertically with an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(x\) satisfies the equation \(300 x ^ { 2 } - 49 x - 255 = 0\).
2. Calculate the value of \(x\).
3. Ignoring the possible effect of air resistance, make

8 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).

1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 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 below \(\boldsymbol { O }\).
   1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
   2. 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 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.

7

1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
2. A block, of mass 4 kg , is attached to one end of a light elastic string. The string has natural length 2 m and modulus of elasticity 196 N . The other end of the string is attached to a fixed point \(O\).
   1. A second block, of mass 3 kg , is attached to the 4 kg block and the system hangs in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-16_374_291_890_877} Find the extension in the string.
   2. The block of mass 3 kg becomes detached from the 4 kg block and falls to the ground. The 4 kg block now begins to move vertically upwards. Find the extension of the string when the 4 kg block is next at rest.
   3. Find the extension of the string when the speed of the 4 kg block is a maximum.
      (3 marks)
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-18_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-19_2486_1714_221_153}

8 An elastic string has one end attached to a point \(O\) fixed on a rough horizontal surface. The other end of the string is attached to a particle of mass 2 kg . The elastic string has natural length 0.8 metres and modulus of elasticity 32 newtons. The particle is pulled so that it is at the point \(A\), on the surface, 3 metres from the point \(O\).

1. Calculate the elastic potential energy when the particle is at the point \(A\).
2. The particle is released from rest at the point \(A\) and moves in a straight line towards \(O\). The particle is next at rest at the point \(B\). The distance \(A B\) is 5 metres. \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-6_179_1055_877_497} Find the frictional force acting on the particle as it moves along the surface.
3. Show that the particle does not remain at rest at the point \(B\).
4. The particle next comes to rest at a point \(C\) with the string slack. Find the distance \(B C\).
5. Hence, or otherwise, find the total distance travelled by the particle after it is released from the point \(A\).

8

1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
2. A particle, of mass 5 kg , is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(O\). The string has natural length 1.6 m and modulus of elasticity 392 N .
   1. Find the extension of the string when the particle hangs in equilibrium.
   2. The particle is pulled down to a point \(A\), which is 2.2 m below the point \(O\). Calculate the elastic potential energy in the string.
   3. The particle is released when it is at rest at the point \(A\). Calculate the distance of the particle from the point \(A\) when its speed first reaches \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 An elastic string has one end attached to a point \(O\), fixed on a horizontal table. The other end of the string is attached to a particle of mass 5 kilograms. The elastic string has natural length 2 metres and modulus of elasticity 200 newtons. The particle is pulled so that it is 2.5 metres from the point \(O\) and it is then released from rest on the table.

1. Calculate the elastic potential energy when the particle is 2.5 m from the point \(O\).
2. If the table is smooth, show that the speed of the particle when the string becomes slack is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The table is, in fact, rough and the coefficient of friction between the particle and the table is 0.4 . Find the speed of the particle when the string becomes slack.

6 When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons. The car has a maximum constant speed of \(48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road.

1. Show that the maximum power of the car is 69120 watts.
2. The car is travelling along a straight horizontal road. Find the maximum possible acceleration of the car when it is travelling at a speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The car starts to descend a hill on a straight road which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the car as it travels on this road down the hill. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-13_2484_1709_223_153} \(7 \quad\) A uniform rod \(A B\), of length 4 m and mass 6 kg , rests in equilibrium with one end, \(A\), on smooth horizontal ground. The rod rests on a rough horizontal peg at the point \(C\), where \(A C\) is 3 m . The rod is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-14_422_984_447_529}

9 A particle, of mass 8 kg , is attached to one end of a length of elastic string. The particle is placed on a smooth horizontal surface. The other end of the elastic string is attached to a point \(O\) fixed on the horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 192 N . \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-18_165_789_571_630} The particle is set in motion on the horizontal surface so that it moves in a circle, centre \(O\), with constant speed \(3 \mathrm {~ms} ^ { - 1 }\). Find the radius of the circle. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-19_2349_1691_221_153} \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-20_2505_1730_212_139}

9 At a theme park, a light elastic rope is used to bring a carriage to rest at the end of a ride. The carriage has mass 200 kg and is travelling at \(8 \mathrm {~ms} ^ { - 1 }\) when the elastic rope is attached to the carriage as it passes over a point \(O\). The other end of the elastic rope is fixed to the point \(O\). The carriage then moves along a horizontal surface until it is brought to rest. The elastic rope is then detached so that the carriage remains at rest. The elastic rope has natural length 6 m and modulus of elasticity 1800 N . The rope, once taut, remains horizontal throughout the motion.

1. Calculate the elastic potential energy of the rope when the carriage is 10 m from \(O\).
   (3 marks)
2. A student's simple model assumes that there are no resistance forces acting on the carriage so that it is brought to rest by the elastic rope alone. Find the distance of the carriage from \(O\) when it is brought to rest.
3. The student improves the model by also including a constant resistance force of 800 N which acts while the carriage is in motion. Find the distance of the carriage from \(O\) when it is brought to rest.
   (8 marks)

8 Zoë carries out an experiment with a block, which she places on the horizontal surface of an ice rink. She attaches one end of a light elastic string to a fixed point, \(A\), on a vertical wall at the edge of the ice rink at the height of the surface of the ice rink. The block, of mass 0.4 kg , is attached to the other end of the string. The string has natural length 5 m and modulus of elasticity 120 N . The block is modelled as a particle which is placed on the surface of the ice rink at a point \(B\), where \(A B\) is perpendicular to the wall and of length 5.5 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-6_499_1429_813_333} The block is set into motion at the point \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) directly towards the point \(A\). The string remains horizontal throughout the motion.

1. Initially, Zoë assumes that the surface of the ice rink is smooth. Using this assumption, find the speed of the block when it reaches the point \(A\).
2. Zoë now assumes that friction acts on the block. The coefficient of friction between the block and the surface of the ice rink is \(\mu\).
   1. Find, in terms of \(g\) and \(\mu\), the speed of the block when it reaches the point \(A\).
   2. The block rebounds from the wall in the direction of the point \(B\). The speed of the block immediately after the rebound is half of the speed with which it hit the wall. Find \(\mu\) if the block comes to rest just as it reaches the point \(B\).

9 Two particles, \(A\) and \(B\), are connected by a light elastic string that passes through a hole at a point \(O\) in a rough horizontal table. The edges of the hole are smooth. Particle \(A\) has a mass of 8 kg and particle \(B\) has a mass of 3 kg . The elastic string has natural length 3 metres and modulus of elasticity 60 newtons.
Initially, particle \(A\) is held 3.5 metres from the point \(O\) on the surface of the table and particle \(B\) is held at a point 2 metres vertically below \(O\). The coefficient of friction between the table and particle \(A\) is 0.4 .
The two particles are released from rest.

1. 1. Show that initially particle \(A\) moves towards the hole in the table.
   2. Show that initially particle \(B\) also moves towards the hole in the table.
2. Calculate the initial elastic potential energy in the string.
3. Particle \(A\) comes permanently to rest when it has moved 0.46 metres, at which time particle \(B\) is still moving upwards. Calculate the distance that particle \(B\) has moved when it is at rest for the first time.

8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
3. Find the maximum value of \(x\).
4. 1. Find the distance fallen by Carol when her speed is a maximum.
   2. Hence find Carol's maximum speed.

3 A light elastic string has natural length 3 m . One end is attached to a fixed point \(O\) and the other end is attached to a particle of mass 1.6 kg . The particle is released from rest in a position 5 m vertically below \(O\). Air resistance may be neglected.

1. Given that in the subsequent motion the particle just reaches \(O\), show that the modulus of elasticity of the string is 117.6 N .
2. Calculate the speed of the particle when it is 4.5 m below \(O\).

7 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-4_122_1009_265_571} As shown in the diagram, \(A\) and \(B\) are fixed points on a smooth horizontal table, where \(A B = 3 \mathrm {~m}\). A particle \(Q\) of mass 1.2 kg is attached to \(A\) by a light elastic string of natural length 1 m and modulus of elasticity \(180 \mathrm {~N} . Q\) is attached to \(B\) by a light elastic string of natural length 1.2 m and modulus of elasticity 360 N .

1. Verify that when \(Q\) is in equilibrium \(B Q = 1.5 \mathrm {~m}\). \(Q\) is projected towards \(B\) from the equilibrium position with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(Q\) oscillates with simple harmonic motion.
2. Show that the period of the motion is 0.314 s approximately.
3. Show that \(u \leqslant 6\).
4. Given that \(u = 6\), find the time taken for \(Q\) to move from the equilibrium position to a position 1.3 m from \(A\) for the first time.