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प्रश्न
Give an example of a physical quantity which has a unit but no dimensions.
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उत्तर
Solid angle Ω = `A/r^2` steradian and a plane angle θ = `L/r` radian. Both are dimensionless but have units.
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संबंधित प्रश्न
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.
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).
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):
If area (A), velocity (V) and density (p) are taken as fundamental units, what is the dimensional formula for force?
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)`
If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity?
- (P – Q)/R
- PQ – R
- PQ/R
- (PR – Q2)/R
- (R + Q)/P
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 neither unit nor dimensions.
If velocity of light c, Planck’s constant h and gravitational contant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.
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.
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.
- Show that the energy equivalent of 1 u is 931.5 MeV.
- 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.
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 ______.
P = `alpha/beta` exp `(-"az"/"K"_"B"theta)`
θ `→` Temperature
P `→` Pressure
KB `→` Boltzmann constant
z `→` Distance
Dimension of β is ______.
