Find the tension in the string \(A P\) and the value of \(\omega\).
Find \(m\) and the tension in the string \(B Q\).
\(6 O\) and \(A\) are fixed points on a rough horizontal surface, with \(O A = 1 \mathrm {~m}\). A particle \(P\) of mass 0.4 kg is projected horizontally with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in the direction \(O A\) and moves in a straight line. After projection, when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between the surface and \(P\) is 0.4 . A force of magnitude \(\frac { 0.8 } { x } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).
Show that, while the particle is in motion, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 - \frac { 2 } { x }\).
It is given that \(P\) comes to instantaneous rest between \(x = 2.0\) and \(x = 2.1\).