Question

A string of length $l$ fixed at one end carries a mass $m$ at the other end. The string makes $\frac{3{2}$ revolutions / second around the vertical axis through the fixed end as shown in figure. The tension $T$ in the string is}

• A. $36 ml$
• B. $3 ml$
• C. $9 ml$
• D. $18 ml$

Hint:

For circular motion, tension provides the centripetal force, which is proportional to the square of angular velocity and the radius.

Answer 1

The Correct Option is A

Step 1: Centripetal force on the mass.
The mass moves in a circle and experiences centripetal force, which is provided by the tension $T$ in the string. The angular velocity $\omega$ is given by:
\[ \omega = 2\pi \times \text{frequency} = 2\pi \times \frac{3}{2} = 3\pi \, \text{rad/s} \]
Step 2: Tension and centripetal force relation.
The centripetal force on the mass is:
\[ T = m \omega^2 l \] Substitute $\omega = 3\pi$:
\[ T = m (3\pi)^2 l = 9 m \pi^2 l \] Since $\pi^2 \approx 3.14$, this simplifies to:
\[ T \approx 36 ml \]
Step 3: Conclusion.
The tension is $36 ml$.