Question

A bob of a simple pendulum of mass $m$ is displaced through $90^\circ$ from its mean position and released. When the bob is at its lowest position, the tension in the string is

• A. $4\ mg$
• B. $2\ mg$
• C. $mg$
• D. $3\ mg$

Hint:

This is a standard scenario in circular motion and mechanics! For any object released from a horizontal position ($90^\circ$), the centripetal force at the bottom is always exactly $2mg$. Adding the rest weight of the object ($1mg$) to counter gravity gives a total vertical tension of exactly $3mg$. Memorizing this relationship can save valuable time during examinations.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Find the tension in a string at the bottommost point when an attached object is released from a horizontal position.

Step 2: Key Formula or Approach:
At the lowest point, total tension T = centripetal force + weight. For release from a horizontal (90°) position, energy conservation gives v² = 2gR, yielding centripetal force mv²/R = 2mg.

Step 3: Detailed Explanation:
The centripetal requirement at the bottom always works out to exactly 2mg for a horizontal release. Adding the object's own weight (mg) to support it against gravity gives a total string tension of 3mg. This 3mg result is a standard benchmark for any object swinging through a quarter-circle arc from rest. Committing this specific value to memory bypasses the full energy and force derivation during timed exams.

Step 4: Final Answer:
The total tension at the bottom is exactly 3mg.