A point mass $m$ is moved in a vertical circle of radius $r$ with the help of a string. The velocity of the ma

Question

A point mass $m$ is moved in a vertical circle of radius $r$ with the help of a string. The velocity of the mass is $\sqrt{7gr}$ at the lowest point. The tension in the string at the lowest point is:

Accepted Answer

1. **Identify Forces:** At the lowest point of the vertical circle, the forces acting on the mass are:
   - Tension ($T$) acting vertically upwards (towards the center).
   - Weight ($mg$) acting vertically downwards.
2. **Apply Newton's Second Law:** The net force towards the center provides the centripetal force ($F_c$) required for circular motion.
   $$ T - mg = \frac{mv^2}{r} $$
   [Source 128]
3. **Substitute Given Values:**
   - Velocity at the lowest point, $v = \sqrt{7gr}$.
   - Radius $= r$.
   $$ T - mg = \frac{m(\sqrt{7gr})^2}{r} $$
   $$ T - mg = \frac{m(7gr)}{r} $$
   $$ T - mg = 7mg $$
4. **Solve for Tension:**
   $$ T = 7mg + mg = 8mg $$
   [Source 50]

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