Circular Motion

A boy whirls a stone in a horizontal circle of radius $r$ and at height $h$ above level ground. The string breaks, and the stone flies off horizontally. It strikes the ground after traveling a horizontal distance of $d$.

An object starts from rest and moves along a circle of radius $R = 3.0$ m with constant tangential acceleration $a_\tau = 3.0$ m/s$^2$.

A horizontal frictionless circular tabletop has radius $R = 1.00$ m and center $O$. A vertical pillar is fixed on the tabletop near its center; the pillar's cross-section at the table surface is a smooth convex closed curve $C$. One end of an inextensible, flexible, thin light string is fixed to a point on the curve $C$, and the other end is tied to a small block of mass $m = 7.5 \times 10^{-2}$ kg. The block is placed on the tabletop with the string taut and is given an initial velocity $v_0 = 4.0$ m/s perpendicular to the string. As the block moves, the string winds around the pillar. The string breaks when its tension reaches $T_0 = 2.0$ N; before breaking, the block always moves on the tabletop and never touches the pillar. Take g = 10 m/s$^2$.