Question

A string of length \( L \) is fixed at one end and carries a mass of \( M \) at the other end. The mass makes \( \frac{3}{\pi} \) rotations per second about the vertical axis passing through the end of the string as shown. The tension in the string is ________________ ML.

Hint:

For rotational motion, the tension in the string acting on a rotating mass is related to the square of the angular velocity and the length of the string. The formula \( T = M \cdot \omega^2 \cdot L \) helps relate the tension to the rotational speed.

Answer 1

Step 1: Problem Setup

A mass \( M \) is attached to a string of length \( L \) and rotates at a frequency of \( \frac{3}{\pi} \) rotations per second. The objective is to determine the tension \( T \) in the string. The mass follows a circular path of radius \( R \), with the string forming an angle \( \theta \) with the vertical.

Step 2: Calculate Angular Velocity

The angular velocity \( \omega \) is derived from the given frequency:

\( \omega = 2\pi \times \text{frequency} = 2\pi \times \frac{3}{\pi} = 6 \, \text{rad/s} \)

Step 3: Determine Centripetal Force

The centripetal force required for the circular motion of mass \( M \) with radius \( R \) is:

\( F_{\text{centripetal}} = M \omega^2 R \)

Substituting \( \omega = 6 \, \text{rad/s} \):

\( F_{\text{centripetal}} = M \times 6^2 \times R = 36 M R \)

Step 4: Analyze Tension Components

The tension \( T \) is resolved into vertical and horizontal components. The vertical component balances gravity:

\( T \cos \theta = Mg \)

The horizontal component equals the centripetal force:

\( T \sin \theta = M \omega^2 R = 36 M R \)

Step 5: Solve for the Angle

The ratio of the horizontal to vertical components of tension yields:

\( \frac{T \sin \theta}{T \cos \theta} = \frac{36 M R}{Mg} \)

This simplifies to:

\( \tan \theta = \frac{36 R}{g} \)

Step 6: Derive Final Tension Expression

The tension \( T \) in the string is calculated as:

\( T = 36 M L \)

Conclusion

The tension in the string is \( \mathbf{36 M L} \).