A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion:
$left parenthesis straight a right parenthesis space straight y space equals space straight a space sin space 2 straight pi space straight t divided by straight T space left parenthesis straight b right parenthesis space straight y space equals space straight a space sin space vt space left parenthesis straight c right parenthesis space straight y space equals space left parenthesis straight a divided by straight T right parenthesis space sin space straight t divided by straight a straight d right parenthesis space straight y space equals space open parentheses straight a square root of 2 close parentheses space left parenthesis sin space 2 space πt divided by straight T space plus space cos space 2 πt space divided by straight T space right parenthesis$
(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 displacement y has the dimension of length.

Therefore, the formula for it should also have the dimension of length.

Trigonometric functions are dimensionless and their arguments are also dimensionless.

Based on these considerations now checking each formula dimensionally, we have

straight a right parenthesis thin space Dimensionally space correct straight y space equals space straight a space sin space fraction numerator 2 πt over denominator straight T end fraction Dimension space of space straight y space equals space straight M to the power of straight o straight L to the power of 1 straight T to the power of straight o space space Dimension space of space straight a space equals space straight M to the power of straight o straight L to the power of 1 straight T to the power of straight o space Dimsension space of space sin space fraction numerator 2 πt over denominator straight T end fraction space equals space straight M to the power of straight o straight L to the power of straight o straight T to the power of straight o space So comma space dimension space of space straight L. straight H. straight S space equals space Dimension space of space straight R. straight H. straight S rightwards double arrow space given space formula space is space dimensionally space correct. straight b right parenthesis thin space Incorrect space dimensionally. straight y space equals space straight a space sin space vt space Dimension space of space straight y space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space space Dimension space of space straight a space equals space straight M to the power of 0 space end exponent straight L to the power of 1 space straight T to the power of 0 space space Dimension space of space vt space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of – 1 end exponent space cross times space straight M to the power of 0 space end exponent straight L to the power of 0 space end exponent straight T to the power of 1 space end exponent equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space The space argument space of space the space trigonometric space function space must space be space dimensionless comma space which space is space not space so space in space this space case. space Hence comma space given space formula space is space dimensionally space incorrect. straight c right parenthesis space Incorrect Dimension space of space straight y space equals space straight M to the power of 0 space space straight L to the power of 1 space space straight T to the power of 0 Dimension space of space begin inline style straight a over straight T end style equals space straight M to the power of 0 space end exponent space straight L to the power of 1 space space straight T to the power of – 1 end exponent space Dimension space of space straight t over straight a equals space equals space straight M 0 space straight L to the power of – 1 space end exponent straight T to the power of 1 space Argument space of space the space trignometric space function space is space dimensionless. space Hence comma space the space formula space is space dimensionally space incorrect. straight d right parenthesis thin space Correct Dimension space of space straight y space equals space straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 Dimension space of space straight a space equals straight M to the power of 0 space straight L to the power of 1 space end exponent straight T to the power of 0 space Dimension space of space straight t over straight T space equals space straight M to the power of 0 space end exponent straight L to the power of 0 space end exponent straight T to the power of 0 space Since space the space srgument space of space the space trignometric space function space is space dimensionless comma space the space dimensions space of space straight y space and space straight a space are space the space same. space Hence comma space the given space formula space is space dimensionally space correct. space

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

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