Advertisements
Advertisements
Question
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.
Solution
By Kepler's third law,
`T^2 ∝ r^3` ⇒ `T ∝ r^(3/2)`
We know that T is a function of R and g.
Let `T ∝ r^(3/2) R^a g^b`
⇒ `T = kr^(3/2) R^a g^b` ......(i)
Where k is a dimensionless constant of proportionality.
Substituting the dimensions of each term in equation (i), we get
`[M^0L^0T] = k[L]^(3/2) [L]^a [LT^-2]^b`
= `k[L^(a+ b + 3/2 T^-2b)]`
On comparing the powers of same terms, we get
`a + b + 3/2` = 0 ......(ii)
`- a2b` = 1 ⇒ b = `- 1/2` ......(iii)
From equation (ii), we get
`a - 1/2 + 3/2` = 0 ⇒ a = – 1
Substituting the values of a and b in equation (i), we get
`T = kr^(3/2) R^-1 g^(-1/2)`
⇒ `T = k/R sqrt(r^3/g)`
APPEARS IN
RELATED QUESTIONS
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.
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?
A physical quantity of the dimensions of length that can be formed out of c, G and `e^2/(4piε_0)` is (c is velocity of light, G is universal constant of gravitation and e is charge):
The dimensional formula for latent heat is ______.
On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct ______.
- y = `a sin (2πt)/T`
- y = `a sin vt`
- y = `a/T sin (t/a)`
- y = `asqrt(2) (sin (2pit)/T - cos (2pit)/T)`
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?
Why length, mass and time are chosen as base quantities in mechanics?
Give an example of a physical quantity which has a unit but no dimensions.
The entropy of any system is given by `S = alpha^2betaIn[(mukR)/(Jbeta^2) + 3]` Where α and β are the constants µ J, k, and R are no. of moles, the mechanical equivalent of heat, Boltzmann constant, and gas constant respectively. `["take S" = (dQ)/T]`
Choose the incorrect option from the following.
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 ______.
