7. A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal as shown. A light string is attached to \(P\) and makes an angle of \(30 ^ { \circ }\) with the plane. When the tension in this string has magnitude \(k m g , P\) is
\includegraphics[max width=\textwidth, alt={}, center]{38e355b0-9d75-40ad-b450-bd74c5135c7f-2_268_474_1759_1407}
just on the point of moving up the plane.

1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac { k \sqrt { } 3 - 1 } { \sqrt { } 3 - k }\).
2. Given further that \(k = \frac { 3 \sqrt { } 3 } { 7 }\), deduce that \(\mu = \frac { \sqrt { } 3 } { 6 }\). The string is now removed.
3. Determine whether \(P\) will move down the plane and, if it does, find its acceleration.
4. Give a reason why the way in which \(P\) is shown in the diagram might not be consistent with the modelling assumptions that have been made.