Question

The magnitude of the displacement of a particle in a circle of radius "a" with constant angular speed v varies with time t as
(A) 2asinvt (B) 2asinvt/2
(C) 2acosvt (D) 2acosvt/2

Answer 1

Dear student

R = a cos θ i+ a sin θj ------(i)

R is the position vector of particle at any time performing circular motion and θ is angular displacement

ω =θ/t or θ =ω t --------(ii)
from (i) and (ii)
R_t = a cos ω t i+ a sin ω t j
at t=0 R₀=a i
Displacement
R_t-R₀=a(cos ωt-1)i + a sin ω t j
magnitude of displacement= [(a(cos ωt-1))²+(a sin ωt)²]^1/2=[2a²-2a²cos ωt]^1/2 =a[2(1-cos ωt)]^1/2

now , cos ωt = (1-2 cos²(ωt/2))

Hence magnitude =2a[cos²(ωt/2)]^1/2 =2 a cos(ωt/2)


Regards