One end of a light inextensible string of length \(2 r \mathrm {~m}\) is attached to a fixed point \(O\). A particle of mass \(m \mathrm {~kg}\) is attached to the other end \(Q\) of the string, so that it can move in a vertical plane. The string is held taut and horizontal and the particle is projected vertically downwards with a speed \(\sqrt { } ( g r ) \mathrm { ms } ^ { - 1 }\). When the string is vertical it begins to wrap round a small, smooth peg \(X\) at a distance \(r \mathrm {~m}\) vertically below \(O\). The particle continues to move.
Find the speed of the particle when it reaches \(O\), in terms of \(g\) and \(r\).
Show that, when \(Q X\) is horizontal, the tension in the string is 3 mgN .
A particle moving along the \(x\)-axis describes simple harmonic motion about the origin \(O\). The period of its motion is \(\frac { \pi } { 2 }\) seconds. When it is at a distance 1 m from \(O\), its speed is \(3 \mathrm {~ms} ^ { - 1 }\). Calculate
the amplitude of its motion,
the maximum acceleration of the particle,
the least time that it takes to move from \(O\) to a point 0.25 m from \(O\).
A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length \(8 l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The two ends of the string are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(A B = 81 \mathrm {~m} . P\) is released from rest at the mid-point of \(A B\).
If \(P\) comes to instantaneous rest at a depth \(3 / \mathrm { m }\) below \(A B\), find an expression for \(\lambda\) in terms of \(m\) and \(g\).
Using this value of \(\lambda\), show that the speed \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) when it passes through the point \(2 l \mathrm {~m}\) below \(A B\) is given by \(v ^ { 2 } = 4 ( 24 \sqrt { 5 } - 53 ) g l\).
A particle \(P\) of mass 0.8 kg moves along a straight line \(O L\) and is acted on by a resistive force of magnitude \(R \mathrm {~N}\) directed towards the fixed point \(O\). When the displacement of \(P\) from \(O\) is \(x \mathrm {~m} , R = \frac { 0 \cdot 8 x v ^ { 2 } } { 1 + x ^ { 2 } }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) at that instant.
Write down a differential equation for the motion of \(P\).
Given that \(v = 2\) when \(x = 0\),
find the speed with which \(P\) passes through the point \(A\), where \(O A = 1 \mathrm {~m}\).
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