A book with many printing errors contains four different formulae for

1.	A book with many printing errors contains four different formulae for the displacement y of a particle undergoing a certain periodic function:(i) y = a sin \(\frac{2\pi t}{T}\)(ii) y = a sin v t(iii) y = \(\frac{a}{T}\) sin \(\frac{t}{a}\)(iv) y = \(\frac{a}{\sqrt{2}}[sin \,\frac{2\pi t}{T}+cos\,\frac{2\pi t}{T}]\)Here, a is maximum displacement of particle, y is speed of particle, T is time period of motion.Rule out the wrong formulae on dimensional grounds.
Answer» The argument of trigonometrical function, i.e., angle is dimensionless. Now, (i) The argument, \([\frac{2\pi t}{T}]=\frac{[T]}{[T]}\) = 1 = [L⁰M⁰T⁰] which is a dimensionless quantity. Hence, formula (i) is correct. (ii) The argument, [vt] = [LT^-1] [T] = [L] = [L¹M⁰T⁰] which is not a dimensionless quantity. Hence, formula (ii) is incorrect. (iii) The argument, \([\frac{t}{a}]=\frac{[T]}{[L]}\) = [L^-1M⁰T¹] which is not a dimensionless quantity. Hence, formula (iii) is incorrect. (iv) The argument, \([\frac{2\pi t}{T}]=\frac{[T]}{[T]}\) = 1 = [L⁰M⁰T⁰] which is a dimensionless quantity. Hence, formula (iv) is correct.

A book with many printing errors contains four different formulae for the displacement y of a particle undergoing a certain periodic function:(i) y = a sin \(\frac{2\pi t}{T}\)(ii) y = a sin v t(iii) y = \(\frac{a}{T}\) sin \(\frac{t}{a}\)(iv) y = \(\frac{a}{\sqrt{2}}[sin \,\frac{2\pi t}{T}+cos\,\frac{2\pi t}{T}]\)Here, a is maximum displacement of particle, y is speed of particle, T is time period of motion.Rule out the wrong formulae on dimensional grounds.

Answer»

The argument of trigonometrical function, i.e., angle is dimensionless.

Now,

(i) The argument, \([\frac{2\pi t}{T}]=\frac{[T]}{[T]}\) = 1 = [L⁰M⁰T⁰]

which is a dimensionless quantity.

Hence, formula (i) is correct.

(ii) The argument,

[vt] = [LT^-1] [T] = [L] = [L¹M⁰T⁰]

which is not a dimensionless quantity.

Hence, formula (ii) is incorrect.

(iii) The argument,

\([\frac{t}{a}]=\frac{[T]}{[L]}\) = [L^-1M⁰T¹]

which is not a dimensionless quantity.

Hence, formula (iii) is incorrect.

(iv) The argument,

\([\frac{2\pi t}{T}]=\frac{[T]}{[T]}\) = 1 = [L⁰M⁰T⁰]

which is a dimensionless quantity.

Hence, formula (iv) is correct.