6.05b Circular motion: v=r*omega and a=v^2/r

4
Two coplanar discs, of radii 0.5 m and 0.3 m , rotate about their centres \(A\) and \(B\) respectively, where \(A B = 0.8 \mathrm {~m}\). At time \(t\) seconds the angular speed of the larger disc is \(\frac { 1 } { 2 } t \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). There is no slipping at the point of contact. For the instant when \(t = 2\), find

1. the angular speed of the smaller disc,
2. the magnitude of the acceleration of a point \(P\) on the circumference of the larger disc, and the angle between the direction of this acceleration and \(P A\).

1 A particle \(P\) is moving in a circle of radius 0.25 m . At time \(t\) seconds, its velocity is \(\left( 2 t ^ { 2 } - 4 t + 3 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). At a particular time \(T\) seconds, where \(T > 0\), the magnitude of the transverse component of the acceleration of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the magnitude of the radial component of the acceleration of \(P\) at this instant.

One end of a light inextensible string of length \(\frac { 3 } { 2 } a\) is attached to a fixed point \(O\) on a horizontal surface. The other end of the string is attached to a particle \(P\) of mass \(m\). The string passes over a small fixed smooth peg \(A\) which is at a distance \(a\) vertically above \(O\). The system is in equilibrium with \(P\) hanging vertically below \(A\) and the string taut. The particle is projected horizontally with speed \(u\) (see diagram). When \(P\) is at the same horizontal level as \(A\), the tension in the string is \(T\). Show that \(T = \frac { 2 m } { a } \left( u ^ { 2 } - a g \right)\). The ratio of the tensions in the string immediately before, and immediately after, the string loses contact with the peg is \(5 : 1\).

1. Show that \(u ^ { 2 } = 5 a g\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is next at the same horizontal level as \(A\).

1 A particle is moving in a circle of radius 2 m . At time \(t \mathrm {~s}\) its velocity is \(\left( t ^ { 2 } - 12 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the magnitude of the resultant acceleration of the particle when \(t = 4\).

7 \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-4_782_1006_274_571} One end of a light inextensible string of length 0.8 m is attached to a fixed point \(A\) which lies above a smooth horizontal table. The other end of the string is attached to a particle \(P\), of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram).

1. Calculate the tension in the string.
2. Calculate the normal contact force between the particle and the table. The particle now moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is on the point of leaving the surface of the table.
3. Calculate \(v\).

5 \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-3_729_739_868_703} A particle \(P\) of mass 0.2 kg is attached to one end of each of two light inextensible strings, one of length 0.4 m and one of length 0.3 m . The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The particle moves in a horizontal circle of radius 0.24 m at a constant angular speed of \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Both strings are taut, the tension in \(A P\) is \(S \mathrm {~N}\) and the tension in \(B P\) is \(T \mathrm {~N}\).

1. By resolving vertically, show that \(4 S = 3 T + 9.8\).
2. Find another equation connecting \(S\) and \(T\) and hence calculate the tensions, correct to 1 decimal place. \section*{[Questions 6 and 7 are printed overleaf.]}

7 \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.2 kg is moving on the smooth inner surface of a fixed hollow hemisphere which has centre \(O\) and radius \(5 \mathrm {~m} . P\) moves with constant angular speed \(\omega\) in a horizontal circle at a vertical distance of 3 m below the level of \(O\) (see Fig.1).

1. Calculate the magnitude of the force exerted by the hemisphere on \(P\).
2. Calculate \(\omega\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_592_773_1231_687} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string is now attached to \(P\). The string passes through a small smooth hole at the lowest point of the hemisphere and a particle of mass 0.1 kg hangs in equilibrium at the end of the string. \(P\) moves in the same horizontal circle as before (see Fig. 2).
3. Calculate the new angular speed of \(P\).

1 \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_378_471_260_836} A uniform square frame \(A B C D\) has sides of length 0.6 m . The side \(A D\) is removed from the frame, and the open frame \(A B C D\) is attached at \(A\) to a fixed point (see diagram).

1. Calculate the distance of the centre of mass of the open frame from \(A\). The open frame rotates about \(A\) in the plane \(A B C D\) with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Calculate the speed of the centre of mass of the open frame.

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
18J
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.
6

1. Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
   6
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
   7
   1. Find the speed of the rocket when the spring first reaches its natural length.
      7
   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 \(e\).
      8
   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]
      8
   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.
      	\(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

1 A child, of mass 40 kg , moves at constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a fairground ride.
The path of the child is a circle of radius 4 metres.
Find the magnitude of the resultant force acting on the child.
Circle your answer.
[0pt] [1 mark]
6.3 N
50 N
130 N
250 N

6 Two particles \(A\) and \(B\) are connected by a light inextensible string. Particle \(A\) has mass 1.2 kg and moves on a smooth horizontal table in a circular path of radius 0.6 m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) moves in a horizontal circle in such a way that it is always vertically below \(A\). The angle that the portion of the string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos \theta = \frac { 4 } { 5 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-4_519_803_484_632}

1. Find the time taken for the particles to perform a complete revolution.
2. Find the mass of \(B\). \section*{END OF QUESTION PAPER}

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 \(O\).

7 Two identical light, inextensible strings \(S _ { 1 }\) and \(S _ { 2 }\) are each of length 5 m . Two identical particles \(P\) and \(Q\) are each of mass 1.5 kg . One end of \(S _ { 1 }\) is attached to \(P\). The other end of \(S _ { 1 }\) is attached to a fixed point \(A\) on a smooth horizontal plane. \(P\) moves with constant speed in a horizontal circular path with \(A\) as its centre (see Fig. 1). One end of \(S _ { 2 }\) is attached to \(Q\). The other end of \(S _ { 2 }\) is attached to a fixed point \(B\). \(Q\) moves with constant speed in a horizontal circular path around a point \(O\) which is vertically below \(B\). At any instant, \(B Q\) makes an angle of \(\theta\) with the downward vertical through \(B\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Given that the angular speed of \(P\) is the same as the angular speed of \(Q\), show that the tensions in \(S _ { 1 }\) and \(S _ { 2 }\) have the same magnitude.
2. You are given instead that the kinetic energy of \(P\) is 39.2 J and that this is the same as the kinetic energy of \(Q\). Determine the difference between the times taken by \(P\) and \(Q\) to complete one revolution. Give your answer in an exact form.

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 1.8 m . The other end of the string is attached to a fixed point, \(O\), on a smooth horizontal plane. Initially, \(P\) is moving with a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) in a horizontal circle with \(O\) as its centre.

1. 1. Find the magnitude of the acceleration of \(P\).
   2. State the direction of the acceleration of \(P\). A force is now applied to \(P\) in such a way that its angular velocity increases. At the instant that the angular velocity reaches \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\), the string breaks.
2. 1. Find the speed with which \(P\) is moving at the instant that the string breaks.
   2. Find the tension in the string at the instant that the string breaks. After the string has broken \(P\) starts to move directly up a smooth slope which is fixed to the plane and inclined at an angle \(\theta ^ { \circ }\) above the horizontal. Particle \(P\) moves a distance of 20 m up the slope before coming to instantaneous rest.
3. Use an energy method to determine the value of \(\theta\).

1 One end of a light inextensible string of length 2.8 m is attached to a fixed point \(O\) on a smooth horizontal table. The other end of the string is attached to a particle \(P\) which moves on the table, with the string taut, in a circular path around \(O\). The speed of \(P\) is constant and \(P\) completes each circle in 0.84 seconds.

1. Find the magnitude of the angular velocity of \(P\).
2. Find the speed of \(P\).
3. Find the magnitude of the acceleration of \(P\).
4. State the direction of the acceleration of \(P\).

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 \(P\).

4 A particle, \(P\), of mass 6 kg is attached to one end of a light inextensible rod of length 2.4 m . The other end of the rod is smoothly hinged at a fixed point \(O\) and the rod is free to rotate in any direction. Initially, \(P\) is at rest, vertically below \(O\), when it is projected horizontally with a speed of \(12 \mathrm {~ms} ^ { - 1 }\). It subsequently describes complete vertical circles with \(O\) as the centre. \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_611_517_536_246} The angle that the rod makes with the downward vertical through \(O\) at each instant is denoted by \(\theta\) and \(A\) is the point which \(P\) passes through where \(\theta = 40 ^ { \circ }\) (see diagram).

1. Find the tangential acceleration of \(P\) at \(A\), stating its direction.
2. Determine the radial acceleration of \(P\) at \(A\), stating its direction.
3. Find the magnitude of the force in the rod when \(P\) is at \(A\), stating whether the rod is in tension or compression. The motion is now stopped when \(P\) is at \(A\), and \(P\) is then projected in such a way that it now describes horizontal circles at a constant speed with \(\theta = 40 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_403_524_1877_242}
4. Find the speed of \(P\).
5. Explain why, wherever \(P\) 's motion is initiated from and whatever its initial velocity, it is not possible for \(P\) to describe horizontal circles at constant speed with \(\theta = 90 ^ { \circ }\).

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.

2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-2_344_340_1418_842}

1. Show that the tension in the string is 22.6 N , correct to three significant figures.
2. Find the speed of the particle.

5 Two light inextensible strings, of lengths 0.4 m and 0.2 m , each have one end attached to a particle, \(P\), of mass 4 kg . The other ends of the strings are attached to the points \(A\) and \(B\) respectively. The point \(A\) is vertically above the point \(B\). The particle moves in a horizontal circle, centre \(B\) and radius 0.2 m , at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle and strings are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_396_558_587_735} $$\text { ← } 0.2 \mathrm {~m} \longrightarrow$$

1. Calculate the magnitude of the acceleration of the particle.
2. Show that the tension in string \(P A\) is 45.3 N , correct to three significant figures.
3. Find the tension in string \(P B\).

5

1. A shiny coin is on a rough horizontal turntable at a distance 0.8 m from its centre. The turntable rotates at a constant angular speed. The coefficient of friction between the shiny coin and the turntable is 0.3 . Find the maximum angular speed, in radians per second, at which the turntable can rotate if the shiny coin is not going to slide.
2. The turntable is stopped and the shiny coin is removed. An old coin is placed on the turntable at a distance 0.15 m from its centre. The turntable is made to rotate at a constant angular speed of 45 revolutions per minute.
   1. Find the angular speed of the turntable in radians per second.
   2. The old coin remains in the same position on the turntable. Find the least value of the coefficient of friction between the old coin and the turntable needed to prevent the old coin from sliding.

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}

4 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular. At time \(t\), the particle's position vector, \(\mathbf { r }\), is given by $$\mathbf { r } = 4 \cos 3 t \mathbf { i } - 4 \sin 3 t \mathbf { j }$$

1. Prove that the particle is moving on a circle, which has its centre at the origin.
2. Find an expression for the velocity of the particle at time \(t\).
3. Find an expression for the acceleration of the particle at time \(t\).
4. The acceleration of the particle can be written as $$\mathbf { a } = k \mathbf { r }$$ where \(k\) is a constant. Find the value of \(k\).
5. State the direction of the acceleration of the particle.