Variable force along axis work-energy

1. A particle \(P\) of mass \(m \mathrm {~kg}\) is initially held at rest at the point \(O\) on a smooth inclined plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\)

The particle is released from rest and slides down the plane against a force which acts towards \(O\). The force has magnitude \(\frac { 1 } { 3 } m x ^ { 2 } \mathrm {~N}\), where \(x\) metres is the distance of \(P\) from \(O\).

1. Find the speed of \(P\) when \(x = 2\) The particle first comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\).

7. A particle \(P\) of mass \(\frac { 1 } { 3 } \mathrm {~kg}\) moves along the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { ( x + 1 ) ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 1\) the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at \(x = 8\) the speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(k\).
2. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
   (Total 14 marks)

1. A particle \(P\) of mass 0.5 kg moves on the positive \(x\)-axis under the action of a single force directed towards the origin \(O\). At time \(t\) seconds the distance of \(P\) from \(O\) is \(x\) metres, the magnitude of the force is \(0.375 x ^ { 2 } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

When \(t = 0 , O P = 8 \mathrm {~m}\) and \(P\) is moving towards \(O\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 260 - \frac { 1 } { 2 } \chi ^ { 3 }\).
2. Find the distance of \(P\) from \(O\) at the instant when \(v = 5\).

2. A particle \(P\) of mass 0.4 kg is moving in a straight line through a fixed point \(O\). At time \(t\) seconds after it passes through \(O\), the distance \(O P\) is \(x\) metres and the resultant force acting on \(P\) is of magnitude ( \(5 + 4 \mathrm { e } ^ { - x }\) ) N in the direction \(O P\). When \(x = 1 , P\) is at the point \(A\).

1. Find, correct to 3 significant figures, the work done in moving \(P\) from \(O\) to \(A\). Given that \(P\) passes through \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),
2. find, correct to 3 significant figures, the speed of \(P\) as it passes through \(A\).

1 A particle \(P\) of mass 4.2 kg is free to move along the \(x\)-axis which is horizontal. \(P\) is projected from the origin, \(O\), in the positive \(x\) direction with a speed of \(2 \mathrm {~ms} ^ { - 1 }\). As \(P\) moves between \(O\) and the point \(A\) where \(x = 4\), it is acted upon by a variable force of magnitude \(\left( 12 x - 3 x ^ { 2 } \right) \mathrm { N }\) acting in the direction \(O A\).

1. Calculate the work done by the force as \(P\) moves from \(O\) to \(A\).
2. Hence, assuming that no other force acts on \(P\), calculate the speed of \(P\) at \(A\).