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If force (F), length (L) and time (T) are assumed to be fundamental units, then the dimensional formula of the mass will be

$F L^{- 1} T^{2}$

$F L^{- 1} T^{- 2}$

$F L^{- 1} T^{- 1}$

$F L^{2} T^{2}$

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Explanation

Let $m = K F^{a} L^{b} T^{c}$

Substituting the dimension of

$[F] = [M L T^{- 2}],$ $[C] = [L] and [T] = [T]$

and comparing both sides, we get $m = F L^{- 1} T^{- 2}$

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The dimensions of universal gas constant is

$[M L^{2} T^{- 2} θ^{- 1}]$

$[M^{2} L T^{- 2} θ]$

$[M L^{3} T^{- 1} θ^{- 1}]$

None of these

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Explanation

$∵ R = \frac{P V}{T}$ $= [\frac{M L^{- 1} T^{- 2} \times L^{3}}{θ}] = [M L^{2} T^{- 2} θ^{- 1}]$

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In the relation $y = a \cos (ω t - k x)$ , the dimensional formula for k is

$[M^{0} L^{- 1} T^{- 1}]$

$[M^{0} L T^{- 1}]$

$[M^{0} L^{- 1} T^{0}]$

$[M^{0} L T]$

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Explanation

$[K x]$ = Dimension of $ω t =$ (dimensionless) hence $K = \frac{1}{X} = \frac{1}{L} = [L^{- 1}]$

$∴ [K] = [L^{- 1}]$

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Position of a body with acceleration 'a' is given by $x = K a^{m} t^{n},$ here t is time. Find dimension of m and n.

m = 1, n = 1

m = 1, n = 2

m = 2, n = 1

m = 2, n = 2

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Explanation

As $x = K a^{m} \times t^{n}$

$[M^{0} L T^{0}] = {[L T^{- 2}]}^{m} {[T]}^{n} = [L^{m} T^{- 2 m + n}]$

$∴$ m = 1 and $- 2 m + n = 0$ $\Rightarrow n = 2$ .

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"Pascal-Second" has dimension of

Force

Energy

Pressure

Coefficient of viscosity

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Explanation

$N S m^{- 2} = N m^{- 2} \times S =$ Pascal-second.

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In a system of units if force (F), acceleration (A) and time (T) are taken as fundamental units then the dimensional formula of energy is

FA²T

FAT²

F²AT

FAT

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Explanation

$E = K F^{a} A^{b} T^{c}$

$[M L^{2} T^{- 2}] = {[M L T^{- 2}]}^{a} {[L T^{- 2}]}^{b} {[T]}^{c}$

$[M L^{2} T^{- 2}] = [M^{a} L^{a + b} T^{- 2 a - 2 b + c}]$

$∴ a = 1, a + b = 2 \Rightarrow b = 1$

and $- 2 a - 2 b + c = - 2 \Rightarrow c = 2$

$∴ E = K F A T^{2}$ .

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The ratio of the dimension of Planck's constant and that of moment of inertia is the dimension of

Frequency

Velocity

Angular momentum

Time

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Explanation

$\frac{h}{I} = [\frac{M L^{2} T^{- 1}}{M L^{2}}] = [T^{- 1}]$

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Which of the following group have different dimension

Potential difference, EMF, voltage

Pressure, stress, young's modulus

Heat, energy, work-done

Dipole moment, electric flux, electric field

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Explanation

Dipole moment, electric flux, electric field

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Out of following four dimensional quantities, which one quantity is to be called a dimensional constant

Acceleration due to gravity

Surface tension of water

Weight of a standard kilogram mass

The velocity of light in vacuum

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Explanation

The velocity of light in vacuum

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Density of a liquid in CGS system is 0.625 g/cm³. What is its magnitude in SI system

0.625

0.0625

0.00625

625

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Explanation

CGS SI

$N_{1} U_{1} = N_{2} U_{2}$

$N_{1} [M_{1} L_{1}^{- 3}] = N_{2} [M_{2} L_{2}^{- 3}]$

$∴ N_{2} = N_{1} [\frac{M_{1}}{M_{2}}] \times {[\frac{L_{1}}{L_{2}}]}^{- 3}$ $= 0 .625 [\frac{1 g}{1 k g}] \times {[\frac{1 c m}{1 m}]}^{- 3}$

$= 0 .625 \times 10^{- 3} \times 10^{6} = 625$

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