Two racing cars of masses m₁ and m₂ are moving in circles of radii r₁

Two racing cars of masses m₁ and m₂ are moving in circles of radii r₁ and r₂ respectively. Their speeds are such that each car makes a complete circle in the same time ‘t’. The ratio of angular speed of the first to that of the second car is :

m₁ : m₂
1 : 1
r₁ : r₂
1 : 2
This question was previously asked in
UPSC CAPF – 2015
The correct answer is 1 : 1.
Angular speed (ω) is defined as the rate of change of angular displacement. For an object moving in a circle, the angular speed is given by ω = 2π / T, where T is the time period (time taken to complete one revolution). The problem states that both cars make a complete circle in the same time ‘t’. Therefore, the time period T is the same for both cars.
For the first car, ω₁ = 2π / t.
For the second car, ω₂ = 2π / t.
The ratio of their angular speeds is ω₁ : ω₂ = (2π / t) : (2π / t) = 1 : 1.
The masses (m₁ and m₂) and radii (r₁ and r₂) of the cars are irrelevant for determining the ratio of angular speeds when the time period is given to be the same.
Linear speed (v) is related to angular speed and radius by v = rω. If the angular speeds are the same but the radii are different, the linear speeds will be different. Centripetal acceleration (a_c = rω²) and centripetal force (F_c = m a_c = m rω²) would also depend on the mass and radius even if the angular speed is constant. This question specifically asks for the ratio of angular speed, which is determined solely by the time period in this case.