C

To solve this problem, we analyze the forces acting on the ball as it moves along its path.
Step 1: Understand the System
The ball is suspended from a point A by a thread of length ℓ. The ball is drawn aside until the thread becomes horizontal.
Step 2: Forces Acting on the Ball
When the ball is in motion, two forces act on it: gravitational force (mg) acting downward and the tension (T) in the thread acting along the thread. As the ball swings down, the gravitational force will change the direction of the tension. At the highest point of the swing, the tension would be zero.
Step 3: Setting Conditions for Tension to be Zero
At the highest point in the trajectory, all the energy initially present in the system gets converted into potential energy. The tension becomes zero when the centripetal force is provided entirely by gravity.
Therefore, at the point where the angle is 90 degrees (highest point), the equation should satisfy:
$$ T + mg = \frac{mv^2}{r} $$
If T = 0 at the height where the angle is 90 degrees, we have: $$ mg = \frac{mv^2}{r} $$
Where r is length ℓ in this case. Using conservation of energy, we can calculate the height achieved.
Step 4: Calculate the Maximum Height and Displacement
The ball moves from its lowest point at O (where it is at rest) to its highest point at A. The distance moved above the point O will be: $$ h = ext{total length} - \frac{\ell}{2} = \ell - \frac{\ell}{2} = \frac{\ell}{2} $$
This indicates the ball will rise by an additional height of $$ \frac{\ell}{2} $$ above point O.
Therefore, the answer to the original question regarding the point of suspension, given the initial length ℓ, the ball will pass through the vertical line corresponding to its lowest point, as this is the point of equilibrium under dynamic motion. Thus the correct answer is the point higher than the starting point when translated along the vertical line to give angle to the tension.
Conclusion:
The total displacement and height obtained are correct at point C.