Question

If the velocity of light \(c\), universal gravitational constant \(G\) and Planck's constant \(h\) are chosen as fundamental quantities The dimensions of mass in the new system is :

• A. $\left[h^1 c ^1 G ^{-1}\right]$
• B. $\left[h^{-1 / 2} c^{1 / 2} G^{1 / 2}\right]$
• C. $\left[h^{1 / 2} c^{1 / 2} G^{-1 / 2}\right]$
• D. $\left[h^{1 / 2} c^{-1 / 2} G^1\right]$

Hint:

Dimensional analysis is a powerful tool. When expressing a quantity in terms of new fundamental quantities, use the dimensional formulas of all quantities and equate the exponents of each fundamental dimension.

Answer 1

The Correct Option is C

 To find the dimensions of mass in the new system where the velocity of light \(c\), the universal gravitational constant \(G\), and Planck's constant \(h\) are chosen as fundamental quantities, we need to express the dimension of mass \([M]\) in terms of these quantities.

The dimensions of the fundamental constants are as follows:

• \([c] = [LT^{-1}]\) (Velocity of light)
• \([G] = [M^{-1}L^3T^{-2}]\) (Gravitational constant)
• \([h] = [ML^2T^{-1}]\) (Planck's constant)

We need to find an expression of mass \([M]\) in the new system using the dimensions of \(c\), \(G\), and \(h\). Assume that:

\([M] = [h]^a [c]^b [G]^c\)

Thus, the dimensional equation becomes:

\([M]^1 = [M^aL^{2a}T^{-a}] \cdot [L^bT^{-b}] \cdot [M^{-c}L^{3c}T^{-2c}]\)

By equating the dimensions, we have:

1. For Mass \([M]\): \(1 = a - c\)
2. For Length \([L]\): \(0 = 2a + b + 3c\)
3. For Time \([T]\): \(0 = -a - b - 2c\)

Solving these equations step-by-step:

• From equation 1: \(a = 1 + c\)
• Substitute \(a = 1 + c\) into equation 2: 
  \(0 = 2(1 + c) + b + 3c\)
  \(0 = 2 + 2c + b + 3c\)
  \(b = -2 - 5c\)
• Substitute \(a = 1 + c\) and \(b = -2 - 5c\) into equation 3:
  \(0 = -(1 + c) - (-2 - 5c) - 2c\) 
  \(0 = -1 - c + 2 + 5c - 2c\)
  \(0 = 1 + 2c\)
  \(c = -\frac{1}{2}\)
• Now, substitute \(c = -\frac{1}{2}\) back to find \(a\) and \(b\):
  \(a = 1 + \left(-\frac{1}{2}\right) = \frac{1}{2}\)
  \(b = -2 - 5(-\frac{1}{2}) = -2 + \frac{5}{2} = \frac{1}{2}\)

Thus, the dimension of mass in this system is:

\([M] = [h]^{1/2}[c]^{1/2}[G]^{-1/2}\)

Therefore, the correct answer is:

$\left[h^{1 / 2} c^{1 / 2} G^{-1 / 2}\right]$