Question

A particle of mass m is tied to a light string of length l and rotated along a vertical circular path. What should be the minimum speed at the highest point of its path so that the string does not become slack at any position?

(a) √(2gl)

(b) √gl 

(c) zero 

(d) √(gl/ 2)

Answer 1

Correct Answer is:(b) √gl 

At A, mv₁²/l = T₁ + mg.

For v₁ to be minimum, T₁ = 0.

or v₁ =√gl.

1/2 mv₂² - 1/2 mv₁² = mg x 2l

or mv₂² / l mv₁² / l + 4mg

At B, mv₂² / l = T₂ - mg

or mg + 4mg = T₂ - mg or T₂ = 6mg.

Applying the principle of conservation of energy between A and D,

1/2 mv ₂³ - 1/2 mv₁² = mgl or mv₃² / l = mv₁² / l + 2mg = 3mg = T₃.

The net force on the bob at D = √((3mg)² + (mg)²) =√10mg.