Question

From the time equations of SHM, prove the relation, v+-omegasqrt(A^(2)-x^(2))

Answer 1

we have to prove that, v = ± ω√{A² - x²}

where ω is angular frequency, A is amplitude and x is displacement of particle.

let standard equation of wave is x = Asin(ωt + Φ) .......(1)

we know, velocity is the rate of change of position with respect to time.

so, dx/dt = ωAcos(ωt + Φ)

= ωA√{1 - sin²(ωt + Φ)}

from equation (1) we get,

dx/dt = v = ωA√{1 - (x/A)²}

= ωA√{A² - x²}/A

= ω√{A² - x²} hence proved

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