Units and Dimensions

The de-Broglie wavelength associated with a particle of mass m and energy E is h/√(2mE). The dimensional formula for Planck's constant is :

• [ML²T^-1]

• [ML^-1T^-2]

• [MLT^-2]

• [M²L²T^-2]

Answer explanation

The correct dimensional formula for Planck's constant is [ML^2T^-1] as derived from the de-Broglie wavelength formula.

The dimensional formula of latent heat is :

• [ML²T^-2]

• [ML²T^-2]

• [MLT^-2]

• [M⁰LT^-2]

Answer explanation

The correct dimensional formula for latent heat is [ML^2T^-2].

If ϵ0 is the permittivity of free space and E is the electric field, then ϵ0E² has the dimensions:

• [M^-1L^-3T⁴A²]

• [ML^-1T^-2]

• [M^oL^-2TA]

• [ML^-1T^-2]

Answer explanation

The correct choice is [ML^-1T^-2] because the dimensions of permittivity (ε0) are M^-1L^-3T^4A^2 and the dimensions of electric field (E) are MLT^-2. Multiplying them gives [ML^-1T^-2].

Statement(I) : Planck's constant and angular momentum have the same dimensions.

Statement(II) : Linear momentum and moment of force have the dimensions.

In Light of the above statement, choose the correct answer from the options given below :

• Statement(I) is correct, Statement(II) is incorrect

• Both statements are incorrect

• Both statements are correct

• Statement(I) is incorrect, Statement(II) is correct

Answer explanation

The correct answer is Statement(I) is correct, Statement(II) is incorrect because Planck's constant and angular momentum have the same dimensions, but linear momentum and moment of force do not have the same dimensions.

• [M^-1L⁵T³]

• [M⁶L⁷T⁴]

• [ML²T^-2]

• [M^oL³T^-2]

Answer explanation

The dimensional formula of ab^-1 is [M^-1L^5T^3]. In the given equation, P is pressure, V is volume, a is a constant, v is velocity, b is a constant, R is the gas constant, and T is temperature.

If G be the gravitational constant and u be the energy density then which of the following quantity have the dimensions as that of the √uG :

• pressure gradient per unit mass

• Gravitational potential

• Energy per unit mass

• Force per unit mass

Answer explanation

The quantity 'Force per unit mass' has the dimensions of √uG, making it the correct choice.

Applying the principle of homogeneity of dimensions, determine which one is correct,

where T is time period, G is gravitational constant, M is mass, r is radius of orbit.

• T² = 4π²r³

Answer explanation

Applying homogeneity of dimensions, correct choice is T^2 = 4π^2r^3/GM.

• The dimensions of n/λ is [T]

• The dimensions of n is [LT^-1]

• The diemsions of x is [L]

• The dimensions of nt is [L]

Answer explanation

The dimensions of n/λ is [T] is NOT correct because n/λ has dimensions of [T^-1] which is the reciprocal of time.

The dimensional formula of angular impulse is :

• [ML^-2T^-1]

• [ML²T^-2]

• [MLT^-1]

• [ML²T^-1]

Answer explanation

The correct dimensional formula for angular impulse is [ML^2T^-1]. This represents the units of mass, length squared, and time to the power of -1.

A force is represented by F=ax²+bt^1/2, where x=distance and t=time. The dimensions of b²/a are :

• [ML³T^-3]

• [MLT^-2]

• [ML^-1T^-1]

• [ML²T^-3]

Answer explanation

The dimensions of b^2/a are [ML^3T^-3] because b^2/a = (M^1L^1T^0)^2 / (M^1L^1T^-2) = M^2L^3T^-3