6.05a Angular velocity: definitions

A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l\) m. The speed of \(P\) is 4 m s\(^{-1}\), and the radius of the circle in which it moves is 2l m. Calculate \(l\). [4]

A small ball \(B\) of mass 0.4 kg moves in a horizontal circle with centre \(O\) and radius 0.6 m on a smooth horizontal surface. One end of a light inextensible string is attached to \(B\); the other end of the string is attached to a fixed point 0.45 m vertically above \(O\).

1. Given that the tension in the string is 5 N, calculate the speed of \(B\). [3]
2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\). [4]

A particle \(P\) of mass \(0.1\text{ kg}\) is attached to one end of a light inextensible string of length \(0.5\text{ m}\). The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a circle which has its centre \(O\) on a smooth horizontal surface \(0.3\text{ m}\) below \(A\). The tension in the string has magnitude \(T\text{ N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R\text{ N}\).

1. Given that the speed of \(P\) is \(1.5\text{ m s}^{-1}\), calculate \(T\) and \(R\). [4]
2. Given instead that \(T = R\), calculate the angular speed of \(P\). [4]

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_3} Particles \(A\) and \(B\), of masses \(m\) and \(3m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles with speed \(v\). Particles \(A\) and \(B\) are at the same horizontal level. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram).

1. Show that \(\cos\theta = \frac{1}{3}\). [2]
2. Find an expression for \(v\) in terms of \(a\) and \(g\). [4]

\includegraphics{figure_6} A light inextensible string is threaded through a fixed smooth ring \(R\) which is at a height \(h\) above a smooth horizontal surface. One end of the string is attached to a particle \(A\) of mass \(m\). The other end of the string is attached to a particle \(B\) of mass \(\frac{1}{2}m\). The particle \(A\) moves in a horizontal circle on the surface. The particle \(B\) hangs in equilibrium below the ring and above the surface (see diagram). When \(A\) has constant angular speed \(\omega\), the angle between \(AR\) and \(BR\) is \(\theta\) and the normal reaction between \(A\) and the surface is \(N\). When \(A\) has constant angular speed \(\frac{3}{2}\omega\), the angle between \(AR\) and \(BR\) is \(\alpha\) and the normal reaction between \(A\) and the surface is \(\frac{1}{2}N\).

1. Show that \(\cos \theta = \frac{4}{9}\cos \alpha\). [5]
2. Find \(N\) in terms of \(m\) and \(g\) and find the value of \(\cos \alpha\). [4]

One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The string is taut and makes an angle \(\theta\) with the downward vertical through \(O\), where \(\cos \theta = \frac{2}{3}\). The particle moves in a horizontal circle with speed \(v\). Find \(v\) in terms of \(a\) and \(g\). [4]

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 small bead \(P\), of mass \(m\) kg, can slide on a smooth circular ring, with centre \(O\) and radius \(r\) m, which is fixed in a vertical plane. \(P\) is projected from the lowest point \(L\) of the ring with speed \(\sqrt{(3gr)}\) ms\(^{-1}\). When \(P\) has reached a position such that \(OP\) makes an angle \(\theta\) with the downward vertical, as shown, its speed is \(v\) ms\(^{-1}\). \includegraphics{figure_5}

1. Show that \(v^2 = gr(1 + 2 \cos \theta)\). [5 marks]
2. Show that the magnitude of the reaction \(RN\) of the ring on the bead is given by $$R = mg(1 + 3 \cos \theta).$$ [4 marks]
3. Find the values of \(\cos \theta\) when
   1. \(P\) is instantaneously at rest,
   2. the reaction \(R\) is instantaneously zero. [2 marks]
4. Hence show that the ratio of the heights of \(P\) above \(L\) in cases (i) and (ii) is \(9:8\). [3 marks]

The radius of the Earth is \(R\) m. The force of attraction towards the centre of the Earth experienced by a body of mass \(m\) kg at a distance \(x\) m from the centre is of magnitude \(\frac{km}{x^2}\) N, where \(k\) is a constant.

1. Show that \(k = gR^2\). [1 mark]

Two satellites \(A\) and \(B\), each of mass \(m\) kg, are moving in circular orbits around the Earth at distances \(3R\) m and \(4R\) m respectively from the centre of the Earth. Given that the satellites move in the same plane and that they lie along the same radial line from the centre at any time,

1. show that the ratio of the speed of \(B\) to that of \(A\) is \(4:3\). [2 marks]

If, in addition, the satellites are linked with a taut, straight wire in the same plane and along the same radial line,

1. find, in terms of \(m\) and \(g\), the magnitude of the force in the wire. [8 marks]

A particle of mass \(m\) kg is attached to one end of a light inextensible string of length \(l\) m whose other end is fixed to a point \(O\). The particle is made to move in a vertical circle with centre \(O\), with constant angular velocity \(\omega\) rad s\(^{-1}\). At a certain instant it is in the position shown, where the string makes an angle \(\theta\) radians with the downward vertical through \(O\). \includegraphics{figure_1}

1. Find an expression, in terms of \(m\), \(l\) and \(\omega\), for the kinetic energy of the particle at this instant. [2 marks]
2. Find an expression, in terms of \(m\), \(g\), \(l\) and \(\theta\), for the potential energy of the particle relative to the horizontal plane through the lowest point \(A\). [2 marks]
3. Determine the position of the particle when the rate of increase of its total energy, with respect to time, is a maximum. [3 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 rectangular lamina \(ABCD\) has mass 20 kg and sides of lengths \(AB = 0.6\) m and \(BC = 1.8\) m. It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(AB\).

1. Show that the moment of inertia of the lamina about the axis is 22.2 kg m\(^2\). [3]

\includegraphics{figure_5} The lamina is released from rest with \(BC\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 N m about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).

1. Show that the angular acceleration is zero when \(\cos \theta = 0.25\). [3]
2. Hence find the maximum angular speed of the lamina. [5]

\includegraphics{figure_7} A uniform rod \(AB\) has mass \(m\) and length \(6a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(AC = a\). The angle between \(AB\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(AB\) and \(S\) perpendicular to \(AB\) (see diagram). The rod is released from rest in the position where \(\theta = \frac{1}{4}\pi\). Air resistance may be neglected.

1. Find the angular acceleration of the rod in terms of \(a\), \(g\) and \(\theta\). [4]
2. Show that the angular speed of the rod is \(\sqrt{\frac{2g(1 - 2\cos\theta)}{7a}}\). [3]
3. Find \(R\) and \(S\) in terms of \(m\), \(g\) and \(\theta\). [6]
4. When \(\cos\theta = \frac{1}{3}\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude. [4]

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]

A cyclist travels around a circular track of radius 20 m at a constant speed of \(8 \text{ m s}^{-1}\) Find the angular speed of the cyclist in radians per second. Circle your answer. \(0.2 \text{ rad s}^{-1}\) \quad\quad \(0.4 \text{ rad s}^{-1}\) \quad\quad \(2.5 \text{ rad s}^{-1}\) \quad\quad \(3.2 \text{ rad s}^{-1}\) [1 mark]

A particle moves in a circular path so that at time \(t\) seconds its position vector, \(\mathbf{r}\) metres, is given by $$\mathbf{r} = 4\sin(2t)\mathbf{i} + 4\cos(2t)\mathbf{j}$$ Find the velocity of the particle, in m s\(^{-1}\), when \(t = 0\) Circle your answer. [1 mark] \(8\mathbf{i}\) \quad \(-8\mathbf{j}\) \quad \(8\mathbf{j}\) \quad \(8\mathbf{i} - 8\mathbf{j}\)

A particle of mass 3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is set into motion so that it moves with a constant speed 4 m s\(^{-1}\) in a circular path with radius 0.8 metres on the horizontal surface.

1. Find the acceleration of the particle. [2 marks]
2. Find the tension in the string. [1 mark]
3. Show that the angular speed of the particle is 48 revolutions per minute correct to two significant figures. [2 marks]

A roundabout in a playground can be modeled as a horizontal circular platform with centre \(O\). The roundabout is free to rotate about a vertical axis through \(O\). A child sits without slipping on the roundabout at a horizontal distance of 1.5 m from \(O\) and completes one revolution in 2.4 seconds.

1. Calculate the speed of the child. [3]
2. Find the magnitude and direction of the acceleration of the child. [3]

One end of a light inextensible string of length \(0.8\) m is attached to a particle \(P\) of mass \(m\) kg. The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3\) m s\(^{-1}\) so that it moves in a vertical circular path with centre \(O\) (see diagram). \includegraphics{figure_1} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac{1}{3}\pi\) radians with the downward vertical through \(O\).

1. Show that at this instant the speed of \(P\) is \(4.5\) m s\(^{-1}\). [3]
2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant. [3]
3. Find the magnitude of the tangential acceleration of \(P\) at this instant. [2]