A bird of mass 0.5 kg , flying around a vertical feeding post at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\), banks its wings to move in a horizontal circle of radius 2 m . The aerodynamic lift \(L\) newtons is perpendicular to the bird's wings, as shown.
Modelling the bird as a particle, find, to the nearest degree, the
\includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_303_472_349_1505}
angle that its wings make with the vertical.
A thin elastic string, of modulus \(\lambda \mathrm { N }\) and natural length 20 cm , passes round two small, smooth pegs \(A\) and \(B\) on the same horizontal level to form a closed loop. \(A B = 10 \mathrm {~cm}\). The ends of the string are attached to a weight \(P\) of mass 0.7 kg .
When \(P\) rests in equilibrium, \(A P B\) forms an equilateral triangle.
\includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_346_371_836_1560}
Find the value of \(\lambda\).
State one assumption that you have made about the weight \(P\), explaining how you have used this assumption in your solution.
A particle \(P\) of mass 0.5 kg moves along a straight line. When \(P\) is at a distance \(x \mathrm {~m}\) from a fixed point \(O\) on the line, the force acting on it is directed towards \(O\) and has magnitude \(\frac { 8 } { x ^ { 2 } } \mathrm {~N}\). When \(x = 2\), the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
Find the speed of \(P\) when it is 0.5 m from \(O\).
A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The other end of the string is attached to a fixed point \(O . P\) is released from rest at \(O\) and falls vertically downwards under gravity. The greatest distance below \(O\) reached by \(P\) is \(2 l \mathrm {~m}\).
Show that \(\lambda = 4 m g\).
Find, in terms of \(l\) and \(g\), the speed with which \(P\) passes through the point \(A\), where
$$O A = \frac { 5 l } { 4 } \mathrm {~m} .$$
(6 marks)
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