Dimensional Analysis and Its Applications

A wave is represented by y = a sin(At - Bx + C) where A, B, C are constants and t is in seconds and x is in metre. The Dimensions of A, B, and C are ______.

An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s Third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that `T = k/R sqrt(r^3/g)`. where k is a dimensionless constant and g is acceleration due to gravity.

If area (A), velocity (V) and density (p) are taken as fundamental units, what is the dimensional formula for force?

A function f(θ) is defined as: `f(θ) = 1 - θ + θ^2/(2!) - θ^3/(3!) + θ^4/(4!)` Why is it necessary for q to be a dimensionless quantity?

The unit of length convenient on the atomic scale is known as an angstrom and is denoted by `Å: 1Å = 10^(-10)m`. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m³ of a mole of hydrogen atoms?

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)`

A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m²s^–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α^–1 β^–2 γ² in terms of the new units.

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