A book with many printing errors contains four different formulae for the displacement y of a particle undergoing a certain periodic function
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.
1 Answers
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.