The correct option (D) (1/4) mω2 A2, (1/4) mω2 A2
Explanation:
Average volume over cycle = (1/T) (t)2∫(t)1 f(t) dt
average PE over a cycle = T∫o {(U ∙ dt)/(T∫o dt)}
= (1/T) T∫o (1/2) KA2 sin2 (ωt + θ)dt (1)
average KE over a cycle
= (1/T) T∫o (1/2) KA2 cos2 (ωt + θ) dt (2)
As T∫o sin2(ωt + θ) dt = (1/2)
& T∫o cos2(ωt + θ) dt = (1/2) & ω = √(K/m)
∴ from (1), avg. PE over cycle = [{(1/2) KA2 [(1/2) dt]T0} / T] = (1/4) KA2
= (1/4) mω2 A2
from (2), avg. KE over cycle = [{(1/2) KA2 [(1/2) dt]T0}/T] = (1/4) KA2
= (1/4) mω2 A2
