A clockmaker is attaching an hour hand to a clock that is hanging on a vertical wall. The hour hand has a mass of and a length of and it is attached to the clock through a point that is at a distance of from one of the ends. The clockmaker is attaching the hand so that it points at 12 o’clock. As he is trying to fix it in place, he slightly disturbs it from its position of equilibrium and it starts to rotate. Assuming that the hour hand is uniform and the axle is smooth, find the angular speed of the hour hand after it has turned through an angle of . Give your answer in terms of , , and the acceleration due to gravity .
An airplane needed to change direction from a bearing of to a bearing of . It did this by banking at an angle of to the horizontal. The banking maneuver caused the airplane to travel in a horizontal circular arc so that, after 30 seconds, it was pointing in the right direction. Given that the plane’s speed was 359 km/h throughout the maneuver, find the two possible values of . Give your answers correct to one decimal place. Take .
A particle of mass lies on a rotating rough disk away from its center, where the coefficient of friction between the particle and the disk is . Given that the disk is rotating horizontally at a constant angular speed about its vertical axis and that remains at rest relative to the disk when , find . Consider the acceleration due to gravity to be .