6. A particle \(P\), of mass 3 kg , moves under the action of two constant forces \(( 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(( 3 \mathbf { i } - \mathbf { 5 j } ) \mathbf { N }\).
- Find, in the form ( \(a \mathbf { i } + b \mathbf { j }\) ) N , the resultant force \(\mathbf { F }\) acting on \(P\).
- Find, in degrees to one decimal place, the angle between \(\mathbf { F }\) and \(\mathbf { j }\).
- Find the acceleration of \(P\), giving your answer as a vector.
The initial velocity of \(P\) is \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
- Find, to 3 significant figures, the speed of \(P\) after 2 s .
\section*{7.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{38f4333d-b374-4064-bf12-e1a0bbd72f56-5_312_1000_397_571}
\end{figure}
A ring of mass 0.3 kg is threaded on a fixed, rough horizontal curtain pole. A light inextensible string is attached to the ring. The string and the pole lie in the same vertical plane. The ring is pulled downwards by the string which makes an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) as shown in Fig. 2. The tension in the string is 2.5 N . Given that, in this position, the ring is in limiting equilibrium, - find the coefficient of friction between the ring and the pole.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{38f4333d-b374-4064-bf12-e1a0bbd72f56-5_315_1000_1497_562}
\end{figure}
The direction of the string is now altered so that the ring is pulled upwards. The string lies in the same vertical plane as before and again makes an angle \(\alpha\) with the horizontal, as shown in Fig. 3. The tension in the string is again 2.5 N . - Find the normal reaction exerted by the pole on the ring.
- State whether the ring is in equilibrium in the position shown in Fig. 3, giving a brief justification for your answer. You need make no further detailed calculation of the forces acting.
(2)
\section*{8.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{38f4333d-b374-4064-bf12-e1a0bbd72f56-6_410_712_388_734}
\end{figure}
Two particles \(P\) and \(Q\) have masses \(3 m\) and \(5 m\) respectively. They are connected by a light inextensible string which passes over a small smooth light pulley fixed at the edge of a rough horizontal table. Particle \(P\) lies on the table and particle \(Q\) hangs freely below the pulley, as shown in Fig. 4. The coefficient of friction between \(P\) and the table is 0.6 . The system is released from rest with the string taut. For the period before \(Q\) hits the floor or \(P\) reaches the pulley, - write down an equation of motion for each particle separately,
- find, in terms of \(g\), the acceleration of \(Q\),
- find, in terms of m and \(g\), the tension in the string.
When \(Q\) has moved a distance \(h\), it hits the floor and the string becomes slack. Given that \(P\) remains on the table during the subsequent motion and does not reach the pulley,
- find, in terms of \(h\), the distance moved by \(P\) after the string becomes slack until \(P\) comes to rest.