Ncert Exercises & Solutions

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 $π$ t/T

(b) y = a sin vt

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

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

The argument of a trigonometrical function, i.e. angle is dimensionless. Now using the principle of homogeneity of dimensions.

(a) \frac{2 π t}{T} = \frac{[T]}{[T]} = d i m e n s i o n l e s s

(b) v t = [L T^{- 1}] [T] = [L] = h a v i n g d i m e n s i o n

(c) \frac{t}{a} = \frac{[T]}{[L T^{- 2}]} = [L^{- 1} T^{- 3}] = h a v i n g d i m e n s i o n

(d) \frac{2 π t}{T} = \frac{[T]}{[T]} = d i m e n s i o n l e s s

S o, (b) a n d (c) a r e d i m e n s i o n a l l y w r o n g .