Multiple particles on string

5 \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-3_927_1022_689_559} One end of a light inextensible string of length 1.2 m is attached to a fixed point \(O\) on a smooth horizontal surface. Particles \(P , Q\) and \(R\) are attached to the string so that \(O P = P Q = Q R = 0.4 \mathrm {~m}\). The particles rotate in horizontal circles about \(O\) with constant angular speed \(\omega \operatorname { rads } ^ { - 1 }\) and with \(O , P\), \(Q\) and \(R\) in a straight line (see diagram). \(R\) has mass 0.2 kg , and the tensions in the parts of the string attached to \(Q\) are 6 N and 10 N .

1. Show that \(\omega = 5\).
2. Calculate the mass of \(Q\).
3. Given that the kinetic energy of \(P\) is equal to the kinetic energy of \(R\), calculate the tension in the part of the string attached to \(O\).

2

1. A particle P of mass \(m\) is attached to a fixed point O by a light inextensible string of length \(a\). P is moving without resistance in a complete vertical circle with centre O and radius \(a\). When P is at the highest point of the circle, the tension in the string is \(T _ { 1 }\). When OP makes an angle \(\theta\) with the upward vertical, the tension in the string is \(T _ { 2 }\). Show that $$T _ { 2 } = T _ { 1 } + 3 m g ( 1 - \cos \theta ) .$$
2. The fixed point A is 1.2 m vertically above the fixed point C . A particle Q of mass 0.9 kg is joined to A , to C , and to a particle R of mass 1.5 kg , by three light inextensible strings of lengths \(1.3 \mathrm {~m} , 0.5 \mathrm {~m}\) and 1.8 m respectively. The particle Q moves in a horizontal circle with centre C , and R moves in a horizontal circle at the same constant angular speed as Q , in such a way that \(\mathrm { A } , \mathrm { C } , \mathrm { Q }\) and R are always coplanar. The string QR makes an angle of \(60 ^ { \circ }\) with the downward vertical. This situation is shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-3_579_1191_881_406} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Find the tensions in the strings QR and AQ .
   2. Find the angular speed of the system.
   3. Find the tension in the string CQ .

\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]

\includegraphics{figure_6} A particle \(P\) of mass \(0.05\text{kg}\) is attached to one end of a light inextensible string of length \(1\text{m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04\text{kg}\) 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 \(0.8\text{m}\) with angular speed \(\omega\text{rads}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4\text{m}\) also with angular speed \(\omega\text{rads}^{-1}\). 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]

\includegraphics{figure_6} A particle \(P\) of mass \(0.05 \text{ kg}\) is attached to one end of a light inextensible string of length \(1 \text{ m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04 \text{ kg}\) 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 \(0.8 \text{ m}\) with angular speed \(\omega \text{ rad s}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4 \text{ m}\) also with angular speed \(\omega \text{ rad s}^{-1}\). 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]

\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]