1. Identify the force providing centripetal force

For a ball moving in a horizontal circle of radius $L$, the tension in the string provides the centripetal force:

$T = m\omega^2 L$

At the maximum possible angular velocity, the tension reaches its maximum allowed value:

$T_{\max} = 324\text{ N}$

2. Substitute the given values

Given:

• $m = 0.5\text{ kg}$
• $L = 0.5\text{ m}$
• $T_{\max} = 324\text{ N}$

So,

$324 = (0.5)\omega^2(0.5)$

$324 = 0.25\omega^2$

3. Solve for $\omega$

$\omega^2 = \frac{324}{0.25} = 1296$

$\omega = \sqrt{1296} = 36\text{ rad/s}$

4. Match with the options

• A: $9$
• B: $18$
• C: $27$
• D: $36$

Hence, the correct option is:

$\boxed{\text{D: }36}$

5. Verification with stored answer

Stored correct answer is D, and our derived answer is also D. So they agree.