Period or time for one revolution

3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of \(3.84 \times 10 ^ { 8 }\) metres.
Find the approximate speed of the moon relative to Earth, in metres per second.

14 J
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42 J 3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of \(3.84 \times 10 ^ { 8 }\) metres.
Find the approximate speed of the moon relative to Earth, in metres per second.
4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).
5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) Find the maximum power output of the engine.
Fully justify your answer.
6 The magnitude of the gravitational force \(F\) between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(d\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is a constant.
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1. Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
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2. The lifetime \(t\) of a planet is thought to depend on its mass \(m\), its radius \(r\), the constant \(G\) and a dimensionless constant \(k\) such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where \(a , b\) and \(c\) are constants.
   Determine the values of \(a , b\) and \(c\).
   7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) As part of a competition, Jo-Jo makes a small pop-up rocket.
   It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
   The mass of the rocket is 18 grams and the stiffness constant of the spring is \(60 \mathrm { Nm } ^ { - 1 }\) Initially the spring is compressed by 3 cm
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   1. Find the speed of the rocket when the spring first reaches its natural length.
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   2. By considering energy find the distance that the rocket rises. 7
   3. In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.
      8 Two smooth spheres \(A\) and \(B\) have the same radius and are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(2 m\) and \(m\) respectively.
      Both \(A\) and \(B\) are initially at rest.
      The sphere \(A\) is set in motion directly towards \(B\) with speed \(3 u\) and at the same time \(B\) is set in motion directly towards \(A\) with speed \(2 u\). Subsequently \(A\) and \(B\) collide directly. \(A\) The coefficient of restitution between the spheres is \(e\).
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   4. Show that the speed of \(B\) after the collision is given by $$\frac { 2 u ( 2 + 5 e ) } { 3 }$$ \section*{Question 8 continues on the next page} 8
   5. Given that the direction of the velocity of \(A\) is reversed during the collision, find the range of possible values of \(e\). Fully justify your answer.
      [0pt] [4 marks]
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   6. Given that the magnitude of the impulse that \(A\) exerts on \(B\) is \(\frac { 19 m u } { 3 }\), find the value of \(e\).
      

      Question number	Additional page, if required. Write the question numbers in the left-hand margin.
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5 One end of a light inextensible string of length 3.5 m is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass \(0.45 \mathrm {~kg} . P\) moves with constant speed in a circular path on the plane with the string taut. The string will break if the tension in it exceeds 70 N . Determine the minimum possible time in which \(P\) can describe a complete circle about \(O\).

8 \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-4_670_819_548_621} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

8 \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-4_604_734_1512_667} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

8 \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-4_614_741_1548_662} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle \(P\) moves in horizontal circles about \(O\). The tension in the string is \(4mg\). Find, in terms of \(a\) and \(g\), the time that \(P\) takes to make one complete revolution. [2]

The gravitational attraction \(F\) N between two point masses \(m_1\) kg and \(m_2\) kg at a distance \(x\) m apart is given by \(F = \frac{km_1m_2}{x^2}\), where \(k\) is a constant. Given that a small body of mass \(1\) kg experiences a force of \(g\) N at the surface of the Earth, which has radius \(R\) m and mass \(M\) kg,

1. show that \(k = \frac{gR^2}{M}\). [2 marks]

A small communications satellite of mass \(m\) kg is put into a circular orbit of radius \(r\) m around the Earth. Modelling the Earth as a particle of mass \(M\) kg, and using the value of \(k\) from (a),

1. prove that the period of rotation, \(T\) s, of the satellite is given by \(T = \frac{2\pi}{R}\sqrt{\frac{r^3}{g}}\). [4 marks]

To cover transmission to any point on the Earth, three small satellites \(X\), \(Y\) and \(Z\), each of mass \(m\) kg, are placed in a common circular orbit of radius \(r\) and form an equilateral triangle as shown. \includegraphics{figure_6}

1. Show on a copy of the diagram the direction of the three forces acting on \(X\). [1 mark]
2. State, with a reason, the direction of the resultant force on \(X\). [2 marks]
3. Show that the period of rotation of \(X\) is given by \(T\sqrt{\frac{3M}{2M + m\sqrt{3}}}\) s, where \(T\) s is the period found in (b). [7 marks]

Assume that the earth moves round the sun in a circle of radius \(1.50 \times 10^8\) km at constant speed, with one complete orbit taking 365 days. Given that the mass of the earth is \(5.97 \times 10^{24}\) kg,

1. calculate the magnitude of the force exerted by the sun on the earth, giving your answer in newtons, [5]
2. state the direction in which this force acts. [1]