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

\includegraphics{figure_3} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T\) N. The acceleration of the particle has magnitude \(7.5 \text{ m s}^{-2}\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\). [4]
2. Find the speed of the particle. [2]

A horizontal circular disc rotates with constant angular speed \(9 \text{ rad s}^{-1}\) about its centre \(O\). A particle of mass \(0.05 \text{ kg}\) is placed on the disc at a distance \(0.4 \text{ m}\) from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc. [3]

\includegraphics{figure_3} Particles \(P\) and \(Q\) have masses \(0.8\) kg and \(0.4\) kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha°\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length \(0.3\) m. The string \(BQ\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius \(0.3\) m about the axis through \(A\) and \(B\) with constant angular speed \(5\) rad s\(^{-1}\) (see diagram).

1. By considering the motion of \(Q\), find the tensions in the strings \(PQ\) and \(BQ\). [3]
2. Find the tension in the string \(AP\) and the value of \(\alpha\). [5]

A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m. One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is 7 N, and the bead rotates with angular speed \(ω\) rad s\(^{-1}\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m, calculate \(ω\), and hence find the kinetic energy of \(B\). [5]
2. Given instead that angle \(ABC = 90°\), and that \(AB\) makes an angle \(\tan^{-1}(\frac{4}{3})\) with the vertical, calculate \(T\) and \(ω\). [6]

\(ABC\) is a uniform semicircular arc with diameter \(AC = 0.5\) m. The arc rotates about a fixed axis through \(A\) and \(C\) with angular speed \(2.4\) rad s\(^{-1}\). Calculate the speed of the centre of mass of the arc. [3]

\includegraphics{figure_4} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45°\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius \(0.67\) m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\). [6]

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]

\includegraphics{figure_5} A light inextensible string \(AB\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(AC = 3a\) and \(DB = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac{3}{4}m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k\omega\). \(AC\) makes an angle \(\theta\) with the downward vertical and \(DB\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\). [7]

A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2m\) is attached to the free end of the string, this particle moves with speed \(\frac{1}{2}v\) in a horizontal circle with centre \(O\) and radius \(\frac{4}{3}x\).

1. Find \(x\) in terms of \(a\). [5]
2. Given that \(v = \sqrt{12ag}\), find the value of \(\lambda\). [2]

Two particles \(A\) and \(B\) of masses \(m\) and \(km\) respectively are connected by a light inextensible string of length \(a\). The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle \(A\) is at a distance \(a\) from the centre of the turntable and particle \(B\) is at a distance \(2a\) from the centre of the turntable. The coefficient of friction between each particle and the turntable is \(\frac{1}{3}\). When the turntable is made to rotate with angular speed \(\frac{2}{5}\sqrt{\frac{g}{a}}\), the system is in limiting equilibrium.

1. Find the tension in the string, in terms of \(m\) and \(g\). [4]
2. Find the value of \(k\). [3]

A particle, of mass 0.8 kg, is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(35°\) to the vertical. The centre of this circle is vertically below \(O\), as shown in the diagram. \includegraphics{figure_4} The particle moves in a horizontal circle and completes 20 revolutions each minute.

1. Find the angular speed of the particle in radians per second. [2 marks]
2. Find the tension in the string. [3 marks]
3. Find the radius of the horizontal circle. [4 marks]

\includegraphics{figure_3} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(PQ = 0.8\) m. A small smooth bead \(B\), of mass 0.01 kg, is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius 0.6 m. \(QB\) rotates with constant angular speed \(\omega\) rad s\(^{-1}\) (see diagram).

1. Show that the tension in the string is 0.1225 N. [3]
2. Find \(\omega\). [3]
3. Calculate the kinetic energy of the bead. [2]

\includegraphics{figure_8} A conical shell has radius 6 m and height 8 m. The shell, with its vertex \(V\) downwards, is rotating about its vertical axis. A particle, of mass 0.4 kg, is in contact with the rough inner surface of the shell. The particle is 4 m above the level of \(V\) (see diagram). The particle and shell rotate with the same constant angular speed. The coefficient of friction between the particle and the shell is \(\mu\).

1. The frictional force on the particle is \(F\) N, and the normal force of the shell on the particle is \(R\) N. It is given that the speed of the particle is 4.5 ms\(^{-1}\), which is the smallest possible speed for the particle not to slip.
   1. By resolving vertically, show that \(4F + 3R = 19.6\). [2]
   2. By finding another equation connecting \(F\) and \(R\), find the values of \(F\) and \(R\) and show that \(\mu = 0.336\), correct to 3 significant figures. [6]
2. Find the largest possible angular speed of the shell for which the particle does not slip. [6]

A bird of mass 0.5 kg, flying around a vertical feeding post at a constant speed of 6 ms\(^{-1}\), banks its wings to move in a horizontal circle of radius 2 m. The aerodynamic lift \(L\) newtons is perpendicular to the bird's wings, as shown. \includegraphics{figure_1} Modelling the bird as a particle, find, to the nearest degree, the angle that its wings make with the vertical. [7 marks]

A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light inextensible string of length 40 cm, as shown. The other end of the string is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\omega\) rad s\(^{-1}\). \includegraphics{figure_2} If the angle \(\theta\) which the string makes with the vertical must not exceed 60°, calculate the greatest possible value of \(\omega\). [7 marks]

A particle of mass \(0.6\) kg moves in a horizontal circle with constant angular speed \(1.5\) radians per second. If the force directed towards the centre of the circle has magnitude \(0.27\) N, find the radius of the circular path. [3 marks]

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]

A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(mg\) N. The other end of the string is attached to a fixed point \(O\). Given that the string makes an angle of \(60°\) with the vertical,

1. show that \(OP = 3l\) m. [4 marks]
2. Find, in terms of \(l\) and \(g\), the angular speed of \(P\). [4 marks]

The diagram shows a particle \(P\) of mass \(m\) kg moving on the inner surface of a smooth fixed hemispherical bowl of radius \(r\) m which is fixed with its axis vertical. \(P\) moves at a constant speed in a horizontal circle, at a depth \(h\) m below the top of the bowl. \includegraphics{figure_6}

1. Show that the force \(R\) exerted on \(P\) by the bowl has magnitude \(\frac{mgr}{h}\) N. [4 marks]
2. Find, in terms of \(g\), \(h\) and \(r\), the constant speed of \(P\). [4 marks]

The bowl is now inverted and \(P\) moves on the smooth outer surface at a height \(h\) above the plane face under the action of a force of magnitude \(mg\) applied tangentially as shown. The reaction of the surface of the sphere on \(P\) now has magnitude \(S\) N. \includegraphics{figure_6b}

1. Given that \(r = 2h\), prove that \(S < \frac{1}{6}R\). [5 marks]

A particle \(P\), of mass 0·5 kg, rests on the surface of a rough horizontal table. The coefficient of friction between \(P\) and the table is 0·5. \(P\) is connected to a particle \(Q\), of mass 0·2 kg, by a light inextensible string passing through a small smooth hole at a point \(O\) on the table, such that the distance \(OQ\) is 0·4 m. \(Q\) moves in a horizontal circle while \(P\) remains in limiting equilibrium. \includegraphics{figure_5}

1. Calculate the angle \(\theta\) which \(OQ\) makes with the vertical. [4 marks]
2. Show that the speed of \(Q\) is 1·33 ms\(^{-1}\). [3 marks]

The motion is altered so that \(Q\) hangs at rest below \(O\) and \(P\) moves in a horizontal circle on the table with speed 0·84 ms\(^{-1}\), at a constant distance \(r\) m from \(O\) but tending to slip away from \(O\).

1. Find the value of \(r\). [5 marks]

A flywheel takes the form of a uniform disc of mass 8 kg and radius 0.15 m. It rotates freely about an axis passing through its centre and perpendicular to the disc. A couple of constant moment is applied to the flywheel. The flywheel turns through an angle of 75 radians while its angular speed increases from 10 rad s\(^{-1}\) to 25 rad s\(^{-1}\).

1. Find the moment of the couple about the axis. [5]

When the flywheel is rotating with angular speed 25 rad s\(^{-1}\), it locks together with a second flywheel which is mounted on the same axis and is at rest. Immediately afterwards, both flywheels rotate together with the same angular speed 9 rad s\(^{-1}\).

1. Find the moment of inertia of the second flywheel about the axis. [3]

A uniform square lamina, of mass 5 kg and side 0.2 m, is rotating about a fixed vertical axis that is perpendicular to the lamina and that passes through its centre. A couple of constant moment 0.06 N m is applied to the lamina. The lamina turns through an angle of 155 radians while its angular speed increases from 8 rad s\(^{-1}\) to \(\omega\) rad s\(^{-1}\). Find \(\omega\). [4]

A uniform circular pulley, of mass \(4m\) and radius \(r\), is free to rotate about a fixed smooth horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. A light inextensible string passes over the pulley and has a particle of mass \(2m\) attached to one end and a particle of mass \(3m\) attached to the other end. The particles hang with the string vertical and taut on each side of the pulley. The rim of the pulley is sufficiently rough to prevent the string slipping. The system is released from rest.

1. Find the angular acceleration of the pulley. [8]

When the angular speed of the pulley is \(\Omega\), the string breaks and a constant braking couple of magnitude \(G\) is applied to the pulley which brings it to rest.

1. Find an expression for the angle turned through by the pulley from the instant when the string breaks to the instant when the pulley first comes to rest. [4]

In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) A ride in a fairground consists of a hollow vertical cylinder of radius 4.6 metres with a horizontal floor. Stephi, who has mass 50 kilograms, stands inside the cylinder with her back against the curved surface. The cylinder begins to rotate about a vertical axis through the centre of the cylinder. When the cylinder is rotating at a constant angular speed of \(\omega\) radians per second, the magnitude of the normal reaction between Stephi and the curved surface is 980 newtons. The floor is lowered and Stephi remains against the curved surface with her feet above the floor, as shown in the diagram. \includegraphics{figure_4}

1. Explain, with the aid of a force diagram, why the magnitude of the frictional force acting on Stephi is 490 newtons. [2 marks]
2. Find \(\omega\) [3 marks]
3. State one modelling assumption that you have used in this question. Explain the effect of this assumption. [2 marks]

A cyclist in a road race is travelling around a bend on a horizontal circular path of radius 15 metres and is prevented from skidding by a frictional force. The frictional force has a maximum value of 500 newtons. The total mass of the cyclist and his cycle is 75 kg Assume that the cyclist travels at a constant speed.

1. Work out the greatest speed, in km h\(^{-1}\), at which the cyclist can travel around the bend. [4 marks]
2. With reference to the surface of the road, describe one limitation of the model. [1 mark]