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Spinning hoop on a smooth floor A uniform circular hoop of radius a rolls and slides on a perfectly smooth horizontal floor. Find its Lagrangian in terms of the Euler angles, and determine which of the generalised momenta are conserved. [Suppose that G has no horizontal motion.]
Investigate the existence of motions in which the angle between the hoop and the floor is a constant α. Show that Ω, the rate of steady precession, must satisfy the equation
cos αΩ2 - 2nΩ - 2(g/a)cot α = 0,
where n is the constant axial spin. Deduce that, for n ≠ 0, there are two possible rates of precession, a faster one going the ‘same way' as n, and a slower one in the opposite direction.
[These are interesting motions but one would need a very smooth floor to observe them.]
If the speed of the man at A (XA = 5m and YA = 1m) is VA = 9 m/s and the increase of his speed is dVA/dt = 13 m/s2, calculate the magnitude of his acceleration and the slop angle at this point.
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A manometer on the discharge pipe of a pump handling 8.4m^3/min of water shows a pressure of 3.7*10^5 Pa. The vacuum gage on the suction pipe shows a vacuum of 21 cm Hg. The vertical distance between the places of connection of the manometer and the ..
A left-turn lane can be installed only if the street in question is widened. A left-turn lane is necessary if the street has traffic lights with an advanced green. The city cannot afford to widen both streets. How many mutually exclusive projects ..
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a hollow vertical cylinder with rigid walls and a height of h is closed on both ends and is filled with an
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