5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Calculate the greatest speed of the particle during its descent.
- Find the greatest distance of the particle below \(O\).
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The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram). - Find the distance of the centre of mass of the rod from \(A\).
The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal. - Calculate \(P\).
In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium. - Find the coefficient of friction at \(A\).
\(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\). - Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\). - Calculate \(V\) and \(a\).
- Find the direction of motion of \(P\) when it passes through \(A\).