a Merry Go Round rotates from rest with constant angular acceleration Alpha ratio time to rotate first two divisions next 2 Revolution series

Dear Student 

​​​​Given that merry go round starts from rest. So ω0 = 0 rad/s.

For first two revolutions, θ = 2×2π rad

θ = ω.t1 + 1/2 α.t1^2

2×2π = 0×t1 + 1/2 × α × t1^2

t1^2 = 8π/α

t1 = √(8π/α)

For first four revolutions, θ = 4×2π rad

θ = ω.t2 + 1/2 α.t2^2

4×2π = 0×t2 + 1/2 × α × t2^2

t2^2 = 16π/α

t2 = √(16π/α)

Time taken for 3rd and 4th revolution is -

t3 = t2 - t1

t3 = √(16π/α) - √(8π/α)

t3 = (√2-1) √(8π/α)

Now, ratio of time to rotate first 2 revolutions and next 2 revolutions is -

t1/t3 = √(8π/α) / [(√2-1)√(8π/α)]

t1/t3 = 1/(√2-1) × (√2+1)/(√2+1)

t1/t3 = √2+1

Therefore, ratio of time to rotate first 2 revolutions & next 2 revolutions is √2+1.

Regards