Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\).
Given instead that \(F = 10 \sqrt { 2 }\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force.
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Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. Particle \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. Particle \(Q\) lies on a plane \(B C\) which is inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
It is given that \(\theta = 60\), the plane \(B C\) is rough and the coefficient of friction between \(Q\) and the plane \(B C\) is 0.7 . The particles are released from rest.
Determine whether the particles move.
It is given instead that the plane \(B C\) is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \(( \sqrt { 3 } - 1 ) \mathrm { N }\).
Find the magnitude of the acceleration of \(P\) as it moves on the plane, and find the value of \(\theta\).