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

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 dimensionless.

(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!)

Posted by

Safeer PP

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