Question 2 - A-Level Maths

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 $$T _ { 2 } = T _ { 1 } + 3 m g ( 1 - \cos \theta ) .$$
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 .