AIIMS 2000 Physics Circular Motion MCQ Question

To determine the minimum height from which the body must fall to complete the circular track, we can use the principle of conservation of energy and the conditions for circular motion. When the body reaches the top of the semicircular track, the gravitational potential energy must be sufficient to provide the necessary centripetal force. At the top of the circle, the centripetal force required is given by $F_c = \frac{mv^2}{R}$, where $R = \frac{D}{2}$ is the radius of the semicircle. The gravitational force acting on the body at the top is $mg$. For the body to just complete the circle, we set the centripetal force equal to the gravitational force: $\frac{mv^2}{R} = mg$. Simplifying gives us $v^2 = gR = g\frac{D}{2}$. Now, using conservation of energy, the potential energy at the height $h$ is equal to the kinetic energy at the top of the semicircle: $mgh = \frac{1}{2} mv^2 + mgD$. Substituting for $v^2$, we have $mgh = \frac{1}{2} m(g\frac{D}{2}) + mgD$. Simplifying this leads to $h = \frac{5}{4}D$. Therefore, the minimum height required is $\frac{5}{4}D$, which corresponds to option B. The other options do not satisfy the energy and motion conditions necessary for completing the circle.