A book with many printing errors contains four different formulae for the displacement y of a particle undergoing a certain periodic motion. Rule out the wrong formula on dimensional ground. ( a= maximum displacement of the particle, v= speed of the particle, T= time period of motion)

Question

A book with many printing errors contains four different formulae for the displacement y of a particle undergoing a certain periodic motion. Rule out the wrong formula on dimensional ground. ( a= maximum displacement of the particle, v= speed of the particle, T= time period of motion) (A) y=a sin (2 π (t/T)) (B) y=(a/T) sin ((t/a)) (C) y=a sin ((v t/a)) (D) y=a √2( sin (2 π (t/T))+ cos ((2 π t/T)))

Tardigrade Admin · Accepted Answer

The argument of a trigonometric function must be dimensionless. Further [y] is not equal to [(a/T)]

Q. A book with many printing errors contains four different formulae for the displacement $y$ of a particle undergoing a certain periodic motion. Rule out the wrong formula on dimensional ground.
( $a =$ maximum displacement of the particle,
$v =$ speed of the particle, $T =$ time period of motion)

$y = a sin (2 π \frac{t}{T})$

$y = \frac{a}{T} sin (\frac{t}{a})$

$y = a sin (\frac{v t}{a})$

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

The argument of a trigonometric function must
be dimensionless. Further $[y]$ is not equal to $[\frac{a}{T}]$

Q. A book with many printing errors contains four different formulae for the displacement y of a particle undergoing a certain periodic motion. Rule out the wrong formula on dimensional ground. ( a= maximum displacement of the particle, v= speed of the particle, T= time period of motion)

y=asin(2πTt​)

y=Ta​sin(at​)

y=asin(avt​)

y=a2​(sin(2πTt​)+cos(T2πt​))