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

A small coin is placed at a point \(C\) on a rough horizontal turntable, with centre \(O\), as shown in the diagram below. \includegraphics{figure_9} The mass of the coin is 3.6 grams. The distance \(OC\) is 20 cm The turntable rotates about a vertical axis through \(O\), with constant angular speed \(\omega\) radians per second.

1. Draw a diagram to show all the forces acting on the coin. [1 mark]
2. The maximum value of friction is 0.01 newtons and the coin does not slip during the motion. Find the maximum value of \(\omega\) Give your answer to two significant figures. [4 marks]
3. State one modelling assumption you have made to answer part (b). [1 mark]

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]

Two particles \(A\) and \(B\) are connected by a light inextensible string of length \(1.26\) m. Particle \(A\) has a mass of \(1.25\) kg and moves on a smooth horizontal table in a circular path of radius \(0.9\) m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) has a mass of \(2\) kg and moves in a horizontal circle as shown in the diagram. The angle that the portion of string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos\theta = \frac{4}{5}\) (see diagram). \includegraphics{figure_7}

1. Determine the angular speed of \(A\) and the angular speed of \(B\). [5]

At the start of the motion, \(A\), \(O\) and \(B\) all lie in the same vertical plane.

1. Find the first subsequent time when \(A\), \(O\) and \(B\) all lie in the same vertical plane. [2]

\includegraphics{figure_6} A particle \(P\) of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m. The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius 0.8 m with angular speed \(\omega\) rad s\(^{-1}\). The particle \(Q\) moves in a horizontal circle of radius 1.4 m also with angular speed \(\omega\) rad s\(^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).

1. Find the tension in the string \(OP\). [3]
2. Find the value of \(\omega\). [3]
3. Find the value of \(\beta\). [2]

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]

\includegraphics{figure_7} A light inextensible string of length 8 m is threaded through a smooth fixed ring, \(R\), and carries a particle at each end. One particle, \(P\), of mass 0.5 kg is at rest at a distance 3 m below \(R\). The other particle, \(Q\), is rotating in a horizontal circle whose centre coincides with the position of \(P\) (see diagram). Find the angular speed and the mass of \(Q\). [8]