A point moves along th e`x` axis according to the law `x=a sin^(2)(omegat-pi//4)` Find.
`(a)` the amplitude and period oscillations, draw the plot `x(t),`
`(b)` the velocity projection `upsilon_(x)` as a function of the coordination `x`, draw the plot `upsilon_(x)(x)`.

Explanation

From the motion law of particle
`x=asin^(2)(omegat-pi//4)=(a)/(2)[a-cos(2omegat-(pi)/(2))]`
or, `x-(a)/(2)=-(a)/(2)cos (2omegat-(pi)/(2))=-(a)/(2)sin 2 omegat =(a)/(2)sin (2omegat+pi)`
i.e. `x=-(a)/(2)=(a)/(2)sin (2omegat+pi) .......(1)`
Now compairing this equation with the general equaltion of harmonic oscillations `:`
`X=Asin (omega_(0)t+alpha)`
Amplitude, `A=(a)/(2)` and angular frequency , `omega_(0)=2omega`.
Thus the period of one full oscillation, `T=(2pi)/(omega_(0))=(pi)/(omega)`
`(b)` Differentiating Eqn (1) w.r.t. time
`v_(x)=a omega cos (2omegat+pi)` or ` v_(x)^(2)=a^(2)omega^(2)cos^(2)(2omegat+pi)=a^(2)omega^(2)[1-sin^(2)(2omegat+pi)] .......(2)`
From Eqn (1) `(x-(a)/(2))^(2)=(a^(2))/(4)sin ^(2)(2omegat+pi)`
or,` 4(x^(2))/(a^(2))+1-(4x)/(a)=sin^(2)(2omegat+pi)` or `1-sin^(2)(2 omegat+pi)= (4x)/(a)(1-(x)/(a)) ....(3)`
From Eqns `(2)` and (3), `v_(x)=a^(2)omega^(2)(4x)/(a)(1-(x)/(a))=4 omega^(2)x(a-x)`
Plot of `v_(x)(x)` is as shown in the answersheet.