Dimensional Analysis and Its Applications

The volume of a liquid flowing out per second of a pipe of length l and radius r is written by a student as `v = π/8 (pr^4)/(ηl)` where P is the pressure difference between the two ends of the pipe and η is coefficient of viscosity of the liquid having dimensional formula ML^–1 T^–1. Check whether the equation is dimensionally correct.

Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass (m ) to energy (E ) as E = mc², where c is speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in MeV, where 1 MeV= 1.6 × 10^–13 J; the masses are measured in unified atomic mass unit (u) where 1u = 1.67 × 10^–27 kg.

1. Show that the energy equivalent of 1 u is 931.5 MeV.
2. A student writes the relation as 1 u = 931.5 MeV. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.

Explain this common observation clearly : If you look out of the window of a fast moving train, the nearby trees, houses, etc. seem to move rapidly in a direction opposite to the train’s motion, but the distant objects (hill tops, the Moon, the stars etc.) seem to be stationary. (In fact, since you are aware that you are moving, these distant objects seem to move with you).

Give an example of a physical quantity which has a unit but no dimensions.

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 `(2pit)/T`

(b) y = a sin vt

(c) y = `(a/T) sin  t/a`

d) y = `(a/sqrt2) (sin 2πt / T + cos 2πt / 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.

In the expression P = E l² m^–5 G^–2, E, m, l and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.

If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity?

1. (P – Q)/R
2. PQ – R
3. PQ/R
4. (PR – Q²)/R
5. (R + Q)/P

On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct ______.

1. y = `a sin  (2πt)/T`
2. y = `a sin vt`
3. y = `a/T sin (t/a)`
4. y = `asqrt(2) (sin  (2pit)/T - cos  (2pit)/T)`