A cyclist travels on a banked track inclined at \(8 ^ { \circ }\) to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of \(v \mathrm {~ms} ^ { - 1 }\). If there is no sideways frictional force on the cycle, calculate the value of \(v\).
The figure shows a particle \(P\), of mass 0.8 kg , attached to the ends of two light elastic strings. \(A P\) has natural length 20 cm and modulus of elasticity \(\lambda \mathrm { N } . B P\) has natural length 20 cm and modulus of of elasticity \(\mu \mathrm { N } . A\) and \(B\) are fixed to points on the same horizontal
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level so that \(A B = 50 \mathrm {~cm}\). When \(P\) is suspended in equilibrium, \(A P =\) 30 cm and \(B P = 40 \mathrm {~cm}\). Calculate the values of \(\lambda\) and \(\mu\).
Suraiya, whose mass is \(m \mathrm {~kg}\), takes a running jump into a swimming pool so that she begins to swim in a straight line with speed \(0.2 \mathrm {~ms} ^ { - 1 }\). She continues to move in the same straight line, the only force acting on her being a resistance of magnitude \(m v ^ { 2 } \sin \left( \frac { t } { 100 } \right) \mathrm { N }\), where \(v \mathrm {~ms} ^ { - 1 }\) is her speed at time \(t\) seconds after entering the pool and \(0 \leq t \leq 50 \pi\).
Find an expression for \(v\) in terms of \(t\).
Calculate her greatest and least speeds during her motion.
A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation \(y = 1 + \cos x\), as shown.
Show by integration that the centre of mass of the lamina is at a distance \(\frac { \pi ^ { 2 } - 4 } { 2 \pi }\) from the \(y\)-axis.
(9 marks)
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Given that the centre of mass is at a distance 0.75 units from the \(x\)-axis, and that \(P\) is the point \(( 0,2 )\) and \(O\) is the origin \(( 0,0 )\),
find, to the nearest degree, the angle between the line \(O P\) and the vertical when the lamina is freely suspended from \(P\).