String through hole – lower particle also moves in horizontal circle (conical pendulum below)

7 Particles \(P\) and \(Q\), of masses 0.8 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string which passes through a small hole in a smooth horizontal table of negligible thickness. \(P\) moves with constant angular speed \(6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a circular path on the surface of the table.

1. It is given that \(Q\) is stationary and that the part of string attached to \(Q\) is vertical. Calculate the radius of the path of \(P\), and find the speed of \(P\).
2. It is given instead that the part of string attached to \(Q\) is inclined at \(60 ^ { \circ }\) to the vertical, and that \(Q\) moves in a horizontal circular path below the table, also with constant angular speed \(6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Calculate the total length of the string.
   [0pt] [6]

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}

3
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]{75f629e7-969d-43ae-8222-031875ae54ae-02_453_696_1571_552}

1. Find the time taken for the particles to perform a complete revolution.
2. Find the mass of \(B\).

\includegraphics{figure_6} A particle \(P\) of mass \(0.2 \text{ kg}\) is attached to one end of a light inextensible string of length \(0.6 \text{ m}\). The other end of the string is attached to a particle \(Q\) of mass \(0.3 \text{ kg}\). The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length \(0.3 \text{ m}\) and modulus of elasticity \(15 \text{ N}\) joins \(Q\) to a fixed point \(A\) which is \(0.4 \text{ m}\) vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended. [4]
2. Find the distance \(HP\) given that the angular speed of \(P\) is \(8 \text{ rad s}^{-1}\). [5]

\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]

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]

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_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]