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Among the following quantities, the quantity whose dimension is independent of mass and length is

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Important Questions on Units and Measurements

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Stoke's law states that the viscous drag force $F$ experienced by a sphere of radius $a$ a, moving with a speed $v$ through a fluid with coefficient of viscosity $η$ , is given by $F = 6 π η a v$ .

If this fluid is flowing through a cylindrical pipe of radius $r$ , length $l$ and a pressure difference of $P$ across its two ends, then the volume of water $V$ which flows through the pipe in time $t$ can be written as, $\frac{V}{t} = k {(\frac{P}{l})}^{a} η^{b} r^{c}$ , where $k$ is a dimensionless constant. Correct values of $a, b$ and $c$ are-

HARD

Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

If energy

(E)

, velocity

(V)

and time

(T)

are chosen as the fundamental quantities, the dimensional formula of surface tension will be___(For surface tension, Force=Surface tension

\times

length)

EASY

Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

x, v

and

a

denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period

T

, then which of the following do not change with time?

EASY

Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Let

L, R, C

and

V

represent inductance, resistance, capacitance and voltage, respectively. The dimension of

\frac{L}{R C V}

S I

units will be:

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

If electronic charge

e

, electron mass

m

, speed of light in vacuum

c

and Planck's constant

h

are taken as fundamental quantities, the permeability of vacuum

μ_{0}

can be expressed in units of:

EASY

Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Time

(T)

, velocity

(C)

and angular momentum

(h)

are chosen as fundamental quantities instead of mass, length and time. In terms of these, the dimensions of mass would be:

EASY

Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Planck's constant

(h)

, speed of light in a vacuum

(c)

and Newton's gravitational constant

(G)

are three fundamental constants. Which of the following combinations of these has the dimension of length?

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

The dimensions of Stefan-Boltzmann constant

σ

can be written in terms of Planck's constant

h

, Boltzmann constant

k_{B}

and the speed of light c as

σ = h^{α} {(k_{B})}^{β} c^{γ}

. Here

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

A beaker contains a fluid of density

ρ

\frac{k g}{m^{3}}

, specific heat

S \frac{J}{k g^{o} C}

and viscosity

η

. The beaker is filled up to height h. To estimate the rate of heat transfer per unit area

(\frac{\dot{Q}}{A})

by convection when beaker is put on a hot plate, a student proposes that it should depend on

η

(\frac{S Δ θ}{h})

and

(\frac{1}{ρ g})

when

Δ θ

(

{i n}^{o} C

) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for

(\frac{\dot{Q}}{A})

is:

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Using dimensional analysis the resistivity in terms of fundamental constants

h, m_{e}, c, e & ε_{0}

can be expressed as

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Young - Laplace law states that the excess pressure inside a soap bubble of radius

R

is given by

∆ P = \frac{4 σ}{R}

where

σ

is the coefficient of surface tension of the soap. The

E \overset{. .}{o} tv \overset{. .}{o} s

number

E_{0}

is a dimensionless number that is used to describe the shape of bubbles rising through a surrounding fluid. It is a combination of the acceleration due to gravity

(g)

, the density of the surrounding fluid

(ρ)

and a characteristic length scale

(L)

which could be the radius of the bubble. A possible expression for

E_{0}

is:

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

If dimensions of critical velocity,

v_{c}

of a liquid flowing through a tube are expressed as

[η^{x} ρ^{y} r^{z}]

, where,

η, ρ

and

r

are the coefficient of viscosity of liquid, density of liquid and radius of the tube, respectively, then, the values of

x

y

and

z

are given by

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

A physical quantity of the dimensions of length that can be formed out of

c, G

and

\frac{e^{2}}{4 π ε_{0}}

, where [

c

is velocity of light,

G

is universal constant of gravitation,

e

is charge &

ε_{0}

is permittivity of free space]:

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

The density of a material in

SI

units is

128 kg m^{- 3} .

In certain units in which the unit of length is

25 cm

and the unit of mass is

50 g,

the numerical value of density of the material is:

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

The dimensions of the area A of a black hole can be written in terms of the universal gravitational constant G, its mass M and the speed of light c as

A = G^{α} M^{β} c^{7} .

Here-

EASY

Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

A, B, C

, and

D

are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation

A D = C l n (B D)

holds true. Then which of the combination is not a meaningful quantity?

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

In the following

I

refers to current and other symbols have their usual meaning. Choose the option that corresponds to the dimensions of electrical conductivity:

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

A neutron star with magnetic moment of magnitude

m

is spinning with angular velocity

ω

about its magnetic axis. The electromagnetic power P radiated by its is given by

μ_{0}^{x} m^{y} ω^{2} c^{u},

where

μ_{n}

and

c

are the permeability and speed of light in free space, respectively. Then

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Let $[\in_{0}]$ denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then :

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Physics>Mechanics>Units and Measurements>Dimensional Formulae and Equations

Expression for time in terms of

G

(universal gravitational constant),

h

(Planck constant) and

c

(speed of light) is proportional to:

Among the following quantities, the quantity whose dimension is independent of mass and length is

Important Questions on Units and Measurements

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