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

Step 1: Express Mass in Terms of Fundamental Quantities

Let \( M \) be the mass, and let its dimensions in terms of \( h \), \( c \), and \( G \) be given by:

\[ M = h^x c^y G^z \]

Step 2: Write the Dimensional Equation

The dimensions of \( h \) (Planck’s constant) are \([ML^2T^{-1}]\). The dimensions of \( c \) (speed of light) are \([LT^{-1}]\). The dimensions of \( G \) (gravitational constant) are \([M^{-1}L^3T^{-2}]\). Substituting these dimensions into the equation from Step 1, we get:

\[ [M] = [M L^2 T^{-1}]^x [L T^{-1}]^y [M^{-1} L^3 T^{-2}]^z \]

\[ [M^1 L^0 T^0] = [M^x L^{2x} T^{-x}] [L^y T^{-y}] [M^{-z} L^{3z} T^{-2z}] \]

\[ [M^1 L^0 T^0] = [M^{x-z} L^{2x+y+3z} T^{-x-y-2z}] \]

Step 3: Equate the Exponents

Equating the exponents of \( M \), \( L \), and \( T \) on both sides, we get the following system of equations:

• \( x - z = 1 \)
• \( 2x + y + 3z = 0 \)
• \( -x - y - 2z = 0 \)

Step 4: Solve the System of Equations

Adding the second and third equations, we get:

\[ x + z = 0 \]

Also, from the first equation, \( x - z = 1 \). Adding these two equations gives \( 2x = 1 \), so \( x = \frac{1}{2} \). Since \( x + z = 0 \), we have \( z = -\frac{1}{2} \). Substituting \( x \) and \( z \) into the third equation gives:

\[ -\frac{1}{2} - y - 2\left(-\frac{1}{2}\right) = 0 \]

\[ -\frac{1}{2} - y + 1 = 0 \]

\[ y = \frac{1}{2} \]

Thus, \( x = \frac{1}{2} \), \( y = \frac{1}{2} \), and \( z = -\frac{1}{2} \).

Step 5: Write the Dimensions of Mass

Therefore, the dimensions of mass in the new system are:

\[ M = h^{\frac{1}{2}} c^{\frac{1}{2}} G^{-\frac{1}{2}} \]

Conclusion:

The correct answer is option (3).