A point moves along x-axis according to the equation x=Asin2(ωt−π4). Find the amplitude, period and velocity as a function of x.
A
A2,πω,2ω√(A−x)x
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B
3A2,πω,ω√(A−x)x
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C
3A2,πω,3ω√(A−x)x
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D
A2,2πω,2ω√(A−x)x
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Solution
The correct option is AA2,πω,2ω√(A−x)x x=Asin2(ωt−π4) =A(sinωtcosπ4−cosωtsinπ4)2=A12(sinωt−cosωt)2 =A2(1−sin2ωt)....(1)
Maximum displacement occurs when sin2ωt=0 ∴ Amplitude = A2
Period is given by 2ω=2ω=2πT∴T=πω dxdt=A2(2ωcos2ωt)=Aωcos2ωt =Aω√1−sin22ωt =Aω√1−(1−2xA)2 (because from (1), ) 1−2xA=sin2ωt =Aω√(2−2xA)(2xA)= 2Aω√(1−xA)(xA)=2ω√(A−x)x