A particle \(P\) of mass 0.25 kg is moving along the positive \(x\)-axis under the action of a single force. At time \(t\) seconds \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where \(\frac { \mathrm { d } v } { \mathrm {~d} x } = 3\). It is given that \(x = 2\) and \(v = 3\) when \(t = 0\)
- Find the magnitude of the force acting on \(P\) when \(x = 5\)
- Find the value of \(t\) when \(x = 5\)
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\caption{Figure 1}
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A cone of semi-vertical angle \(60 ^ { \circ }\) is fixed with its axis vertical and vertex upwards. A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60 ^ { \circ }\) with the horizontal, as shown in Figure 1.