2.14 A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion :

(a) y = a sin 2\pi t/T

(b) y = a sin vt

(c) y = (a/T) sin\ t/a

(d) y = ( a\sqrt{2}) (sin 2\pi t / T+ cos 2\pi t / T )

(a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional grounds.

Answers (1)

Ground rules:

\\sin\Theta , cos\Theta \ are\ DIMENSIONLESS.\ \\ and\ \Theta\ must\ be\ dimensionless

[y] = L  (M^0 L^1 T^0)

[a] = L 

[v] = LT^{-1}

[t/T] is Dimenionless.

(a) The dimensions on both sides are equal, the formula is dimensionally correct.

(b) \because [vt] = (LT^{-1})(T) = L (\therefore \Theta is not dimensionless)

        The formula is dimensionally incorrect

(c) [a/T] = (L)/(T)

    It is dimensionally incorrect, as the dimensions on both sides are not equal.

(d) The dimensions on both sides are equal, the formula is dimensionally correct. (Don't get confused by summation of trigonometric functions !)

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