A ball of mass $$0.5$$ kg is attached to a string of length $$50$$ cm. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is $$400$$ N. The maximum possible value of angular velocity of the ball in rad s$$^{-1}$$ is :

A ball of mass 0.5 kg on a string of length 50 cm rotates in a horizontal circle. Maximum tension = 400 N. Find maximum angular velocity.

For circular motion, the string tension provides the centripetal force needed to keep the ball moving in a circle:

$$ T = m\omega^2 r $$

where $$T$$ is tension, $$m$$ is mass, $$\omega$$ is angular velocity, and $$r$$ is radius of the circular path.

Rearranging for $$\omega$$ gives:

$$ \omega^2 = \frac{T}{mr} $$

Substituting the values $$T = 400\text{ N}$$, $$m = 0.5\text{ kg}$$, and $$r = 50\text{ cm} = 0.5\text{ m}$$ yields:

$$ \omega^2 = \frac{400}{0.5 \times 0.5} = \frac{400}{0.25} = 1600 $$

$$ \omega = \sqrt{1600} = 40\text{ rad/s} $$

The correct answer is Option B: 40 rad/s.