With what angular velocity should a 20 m long cord be rotated such that tension in it, while reaching the highest point, is zero

The correct option is 0.7 rad/sec.

To find the angular velocity required for the tension in the cord to be zero at the highest point of a vertical circle, we analyze the forces acting on the rotating object at that point.

At the highest point of the vertical circle, both the gravitational force and the tension in the cord act vertically downwards towards the center of the circle. Together, they provide the necessary centripetal force for the circular motion:

$T+mg=m{\omega }^2$
where:
- $T$ is the tension in the cord,
- $m$ is the mass of the object,
- $g$ is the acceleration due to gravity (taken as 9.8 m/s²),
- $L$ is the length of the cord (20 m), which acts as the radius of the circular path,
- $\omega$ is the angular velocity.

We are given that the tension at the highest point is zero ($T=0$). Substituting this value into our force equation, we get:

$0+mg=m{\omega }^2$

$mg=m{\omega }^2$

We can simplify the equation by dividing both sides by the mass ($m$):

$g={\omega }^2$

${\omega }^2=\frac{g}{L}$

$\omega =\sqrt{\frac{g}{L}}$

Now, we substitute the given values, $L=20\text{m}$ and $g=9.8{\text{m/s}}^2$, into the formula:

$\omega =\sqrt{\frac{9.8}{20}}$

$\omega =\sqrt{0.49}$

$\omega =0.7\text{rad/sec}$

Thus, the required angular velocity is 0.7 rad/sec.