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A particle is executing SHM represented by the equation x(t) = Acos⁡(ωt + ϕ). Given that the initial distance of the particle from its equilibrium position is 1 cm and the initial velocity is π cms s−1 and its angular frequency is πs−1. Using this, data, calculate the amplitude of the motion.

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a

πcm

b

2 cm

c

2cm

d

1 cm

answer is C.

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Detailed Solution

x=Acos⁡(ωt+ϕ) where A is amplitudeAt t = 0, x = 1 cm∴1=Acos⁡ϕ…(i)Velocity,v=dxdt=ddt(Acos⁡(ωt+ϕ))=−Aωsin⁡(ωt+ϕ)At t = 0, v=πcms−1∴π=−Aωsin⁡ϕ or πω=−Asin⁡ϕ∵ω=πs−1∴1=−Asinϕ ......iiSquaring and adding (i) and (ii), we getA2cos2⁡ϕ+A2sin2⁡ϕ=2A2=2∵sin2⁡ϕ+cos2⁡ϕ=1∴A=2cm

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A particle is executing SHM represented by the equation x(t) = Acos⁡(ωt + ϕ). Given that the initial distance of the particle from its equilibrium position is 1 cm and the initial velocity is π cms s−1 and its angular frequency is πs−1. Using this, data, calculate the amplitude of the motion.