6.05a Angular velocity: definitions

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]

A particle of mass 1.2 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is fixed to a point O on a smooth horizontal surface. With the string taut, the particle moves on the surface with constant speed \(8\text{ ms}^{-1}\) in a horizontal circle with centre O.

1. Find the angular velocity of the particle about O. [2]
2. Calculate the tension in the string. [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]

\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]