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A book with many printing errors contains four different forumlae for the displacement y of a particle undergoing a certain periodic motion : (i) y = a sin (2pi t)/(T) (ii) y = a sin upsilon t (iii)y = (a)/(T) sin (t)/(a) (iv)y= (a)/(sqrt2)[sin(2pi t)/(T) + cos (2pi t)/(T)] Here, a is maximum displacement of particle, upsilon is speed of particle, T is time period of motion. Rule out the wrong forumlae on dimensinal grounds.

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Answer Text

Solution :
According to dimensional analysis an equation must be dimensionally homoheneous. <br> (a) `y = a "sin" (2pi t)/T` <br> Here, `[L.H.S.]=[y]=[L]` <br> and `[R.H.S.]=[a "sin" (2 pi t)/T]=[L "sin"T/T]=[L]` <br> So, it is correct. <br> (b) `y = a sin vt` <br> Here, `[y]=[L]` and `[a sin vt]=[L sin (LT^(-1).T)]=[L sin L]` <br> so, the equation is wrong. <br> (c) `y = (a/T)"sin" t/a` <br> Here, `[y]=[L]` and `[(a/T)"sin" t/a]=[L/T"sin"T/L]=[LT^(-1) sin TL^(-1)]` <br> So, the equation is wrong. <br> (d) `y=(asqrt(2))("sin"(2pit)/T+"cos"(2pit)/T)` <br> Here, `[y]=[L], [asqrt(2)]=[L]` <br> and `["sin"(2pit)/T+"cos"(2pit)/T]=["sin"T/T+ "cos"T/T]=` dimensionless <br> So, the equation is correct.

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