A particle is hanging from a fixed point O by means of a string of length a. There is a small nail Q in the same horizontal line with O at a distance of a/3 from O. Find the minimum velocity with which the particle should be projected so that it may make a complete revolution around the nail without being slackened.please make the answer more detailed and explained

To solve this problem, we need to determine the minimum velocity required for the particle to complete a full revolution around the nail Q without the string becoming slack.

Step 1: Understanding the Motion
The particle is initially suspended from a fixed point O by a string of length a.
A nail Q is positioned at a distance a/3 from O along the same horizontal line.
The particle is given an initial velocity so that it swings and wraps around the nail, forming a new effective length of 2a/3.
The condition for completing the revolution around Q is that the particle must reach the topmost point with at least zero tension in the string, meaning it must have at least the velocity required for circular motion at that point.
Step 2: Minimum Speed at the Top of the Circular Path
At the topmost point of the new circular motion (with center at Q and radius 2a/3), the forces acting on the particle are:

Tension T in the string (which must be at least zero for the motion to continue).
The gravitational force mg acting downward.
The required centripetal force for circular motion at the topmost point is provided by gravity and tension:

T + mg = (m v²) / r

For the minimum velocity case, T = 0 (just about to become slack), so:

mg = (m v²) / r

Since the new radius r = 2a/3, we get:

mg = (m v²) / (2a/3)

Cancel m from both sides:

g = v² / (2a/3)

Rearranging,

v² = (2a/3) g

So,

v_top = sqrt((2a/3) g)

Step 3: Energy Conservation
We now apply energy conservation between the initial position (lowest point) and the topmost point of the circular path.

At the lowest point:

Initial velocity = u (to be determined).
Kinetic energy = (1/2) m u².
Potential energy = 0 (taking the lowest point as reference).
At the topmost point:

Velocity = v_top = sqrt((2a/3) g).
Kinetic energy = (1/2) m (2a/3 g).
Potential energy = mg (4a/3) (since the particle rises by 4a/3 from the lowest position to the top).
Using energy conservation:

Initial total energy = Final total energy

(1/2) m u² = (1/2) m (2a/3 g) + mg (4a/3)

Cancel m:

(1/2) u² = (1/2) (2a/3 g) + (4a/3) g

(1/2) u² = (a/3 g) + (4a/3 g)

(1/2) u² = (5a/3) g

Multiply both sides by 2:

u² = (10a/3) g

Taking the square root:

u = sqrt((10a/3) g)

Step 4: Final Answer
The minimum velocity required for the particle to complete a full revolution around the nail without slackening is:

u = sqrt((10a/3) g).