A body initially at rest and sliding along a frictionless track from a height $h$ just completes a vertical ci

Question

A body initially at rest and sliding along a frictionless track from a height $h$ just completes a vertical circle of diameter $AB = D$. The height $h$ is equal to:

Accepted Answer

1. **Identify Condition for Vertical Circle:** For a body to just complete a vertical circle, the minimum velocity ($v$) required at the lowest point of the circle is given by $v = \sqrt{5gR}$, where $R$ is the radius [Class 11 Physics, Ch 5, Example 5.7].
2. **Relate Radius to Diameter:** Given the diameter $D$, the radius $R = \frac{D}{2}$. Thus, $v = \sqrt{5g\frac{D}{2}}$.
3. **Apply Conservation of Mechanical Energy:** The body starts from rest at height $h$. The potential energy lost is converted into kinetic energy at the bottom of the circle.
   $$ PE_{initial} = KE_{final} $$
   $$ mgh = \frac{1}{2}mv^2 $$ [Class 11 Physics, Ch 5, Sec 5.8]
4. **Substitute and Solve:**
   $$ mgh = \frac{1}{2}m \left( \sqrt{\frac{5gD}{2}} \right)^2 $$
   $$ gh = \frac{1}{2} \left( \frac{5gD}{2} \right) $$
   $$ h = \frac{5}{4}D $$

Step-by-Step Solution

Practice All PHYSICS Questions