All Problems The Fundamental Equation of Dynamics
Problem 1.108
A small body is placed on the top of a smooth sphere of radius \(R .\) Then the sphere is imparted a constant acceleration \(w_{0}\) in the horizontal direction and the body begins sliding down. Find: (a) the velocity of the body relative to the sphere at the moment of break-off; (b) the angle \(\theta_{0}\) between the vertical and the radius vector drawn from the centre of the sphere to the break-off point; calculate \(\theta_{0}\) for \(w_{0}=g\).
1.108. (a) \(v=\sqrt{2 g R / 3}\); (b) \(\cos \theta_{0}=\frac{2+\eta \sqrt{5+9 \eta^{2}}}{3\left(1+\eta^{2}\right)},\) where \(\eta=\) \(=w_{0} / g, \theta_{0} \approx 17^{\circ}\)