Two bodies A & B rotate about an axis, such that angle θA (in radians) covered by first body is proportional to square of time, & θB (in radians) covered by second body varies linearly. At t = 0, θA = θB = 0. If A completes its first revolution in sec. & B needs 4π sec. to complete half revolution then; angular velocity ωA : ωB at t = 5 sec. are in the ratio
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Given: θA = k t² (proportional to square of time) and θB = c t (varies linearly). At t=0, both are 0.
For A: completes first revolution (2π rad) at t=√π s. So, 2π = k (√π)² ⇒ k = 2.
Thus, θA = 2t². Angular velocity ωA = dθA/dt = 4t. At t=5 s, ωA = 20 rad/s.
For B: completes half revolution (π rad) in 4π s. So, π = c (4π) ⇒ c = 1/4.
Thus, θB = (1/4)t. Angular velocity ωB = dθB/dt = 1/4 rad/s (constant).
Therefore, ωA : ωB = 20 : (1/4) = 80 : 1.
Final Answer: 80 : 1