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Question -

A book with many printing errors contains four different formulasfor the displacement y of a particle undergoing a certainperiodic motion:

(a) 

(b) y = a sin vt

(c)  

(d) 

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



Answer -

(a) Answer: Correct

 

Dimension of y = M0 L1 T0

Dimension of a = M0 L1 T0

Dimensionof = M0 L0 T0

Dimension of L.H.S = Dimension ofR.H.S

Hence, the given formula is dimensionally correct.

(b) Answer: Incorrect

y = a sin vt

Dimension of y = M0 L1 T0

Dimension of a = M0 L1 T0

Dimension of vt = M0 L1 T–1 ×M0 L0 T1 = M0 L1 T0

But the argument of the trigonometric function must bedimensionless, which is not so in the given case. Hence, the given formula isdimensionally incorrect.

(c) Answer: Incorrect

Dimension of y = M0L1T0

Dimensionof = M0L1T–1

Dimensionof= M0 L–1 T1

But the argument of the trigonometric function must bedimensionless, which is not so in the given case. Hence, the formula isdimensionally incorrect.

(d) Answer: Correct

Dimension of y = M0 L1 T0

Dimension of a = M0 L1 T0

Dimensionof  = M0 L0 T0

Since the argument of the trigonometric function must bedimensionless (which is true in the given case), the dimensions of y and a arethe same. Hence, the given formula is dimensionally correct.

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