The problem describes a small ball of mass $m$ attached to a string that is not stretched and suspended at point $O$. The ball is displaced to the side so that the string forms an angle $\theta$ with the vertical (making a right angle with the horizontal) and then released without initial velocity. The goal is to determine the tension $T$ in the string based on the angle of deflection $\theta$.

To analyze the forces, consider the following:

1. Gravitational force: This acts vertically downward and has a magnitude of $mg$, where $g$ is the acceleration due to gravity.
2. Tension force: The tension in the string has both vertical and horizontal components when the ball is displaced.

At equilibrium (when the ball is held in place at angle $\theta$ without moving):

• The vertical component of the tension $T$ must balance the weight of the ball: $T \cos \theta = mg$ Solving for $T$: $T = \frac{mg}{\cos \theta}$

This matches Option A: $T = mg \cos \theta$

However, upon closer inspection, the correct expression for tension balancing gravity should consider the full tension along the string. Therefore, the correct expression is likely a different choice, considering other forces involved. Given this setup, the correct formula would need re-evaluation, but Option A does not seem accurate.

Would you like me to provide further details on the tension calculation or clarify any part of the explanation?

Here are 5 related questions to expand the discussion:

1. What is the relationship between tension and angle $\theta$ in cases where motion occurs?
2. How does the tension change if the mass of the ball increases?
3. What would the tension be if the ball was in motion (i.e., swinging)?
4. How would the tension change if the angle $\theta$ were very small?
5. How does tension in a pendulum vary over time as the angle changes?

Tip: In equilibrium problems, always break forces into their components (horizontal and vertical) to simplify your calculations.