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 T 2 π t (b) y = a sin v t (c) y = ( T a ) sin a t (d) y = ( a 2 ) ( sin T 2 π t + cos T 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.

Question

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 T 2 π t ​ (b) y = a sin v t (c) y = ( T a ​ ) sin a t ​ (d) y = ( a 2 ​ ) ( sin T 2 π t ​ + cos T 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.,

MARKS App · Accepted Answer

According to dimensional analysis an equation must be dimensionally homogeneous. (a) y = a sin T 2 π t ​ Here, [ L.H.S. ] = [ y ] = [ L ] and [ R.H.S. ] = [ a sin T 2 π t ​ ] $ = [ L sin T T ​ ] = [ L ] S o , i t i scorrec t . ( b ) y=a \sin v \mathrm{t} \quad Here , [y]=[\mathrm{L}] an d [a \sin v \mathrm{t}]=\left[\mathrm{L} \sin \left(\mathrm{LT}^{-1} \cdot \mathrm{T}\right)\right] =[\mathrm{L} \sin \mathrm{L}] S o , t h ee q u a t i o ni s w ro n g . ( c ) y=\left(\frac{a}{T}\right) \sin \frac{t}{a} \quad Here , [ y ] =[ L ] an d \left[\left(\frac{a}{\mathrm{~T}}\right) \sin \frac{t}{a}\right]=\left[\frac{\mathrm{L}}{\mathrm{T}} \sin \frac{\mathrm{T}}{\mathrm{L}}\right] = [ LT − 1 sin TL − 1 ] S o , t h ee q u a t i o ni s w ro n g . ( d ) y=(a \sqrt{2})\left(\sin \frac{2 \pi t}{\mathrm{~T}}+\cos \frac{2 \pi t}{\mathrm{~T}}\right) Here , [y]=[\mathrm{L}],[a \sqrt{2}]=[\mathrm{L}] and [ sin T 2 π t ​ + cos T 2 π t ​ ] = [ sin T T ​ + cos T T ​ ] ∵ [ LHS ] = [ RHS ] =$ dimensionless So, the equation is correct.

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