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.
The problem involves a rotating mass \( M \) attached to a string of length \( L \), making \( \frac{3}{\pi} \) rotations per second. The question asks for the tension \( T \) in the string. The mass describes a circular path with a radius \( R \), and the string makes an angle \( \theta \) with the vertical axis.
The number of rotations per second (frequency) is given as \( \frac{3}{\pi} \) rotations per second. The angular velocity \( \omega \) is related to the frequency by:
\( \omega = 2\pi \times \text{frequency} = 2\pi \times \frac{3}{\pi} = 6 \, \text{rad/s} \)
The mass \( M \) is undergoing circular motion with a radius \( R \). The centripetal force required to keep the mass in its circular path is given by:
\( F_{\text{centripetal}} = M \omega^2 R \)
Substituting the value of \( \omega \) (which is 6 rad/s), we get:
\( F_{\text{centripetal}} = M \times 6^2 \times R = 36 M R \)
The tension \( T \) in the string has both vertical and horizontal components. The vertical component of the tension balances the gravitational force acting on the mass:
\( T \cos \theta = Mg \)
The horizontal component provides the centripetal force:
\( T \sin \theta = M \omega^2 R = 36 M R \)
Using the relationship between the vertical and horizontal components of tension:
\( \frac{T \sin \theta}{T \cos \theta} = \frac{36 M R}{Mg} = \frac{36 R}{g} \)
This simplifies to:
\( \tan \theta = \frac{36 R}{g} \)
Finally, the tension \( T \) in the string can be expressed as:
\( T = 36 M L \)
The tension in the string is \( \mathbf{36 M L} \).