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Question

If the constant of gravitation (G), Plank’s constant (h) and the velocity of light (c) be chosen as fundamental units. The dimension of the radius of  gyration is

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Dimensional Analysis
	**Related Topics**: Fundamental Units, Derived Units
	**Prerequisites**: Knowledge of dimensional formulas for G, h, c and radius of gyration.
## SOLUTION PATHWAY
**Given Information**:
	* Fundamental units: G, h, c
	* Need to find the dimension of radius of gyration in terms of G, h, and c.

**Method & Steps**:
	1. **Write the dimensional formulas for the given constants and radius of gyration:**
		* Radius of Gyration (k): [L] or $M^0L^1T^0$
		* Gravitational Constant (G): $M^{-1}L^3T^{-2}$
		* Planck's Constant (h): $ML^2T^{-1}$
		* Speed of Light (c): $LT^{-1}$
		Reasoning: These are standard dimensional formulas.
	2. **Assume the dimension of the radius of gyration (k) is related to G, h, and c as follows:**
		$k = G^a h^b c^d$
		Reasoning: We express the dimension of k as a combination of G, h, and c with unknown powers a, b, and d.
	3. **Equate the dimensions on both sides:**
		$L = (M^{-1}L^3T^{-2})^a (ML^2T^{-1})^b (LT^{-1})^d$
		$M^0L^1T^0 = M^{-a+b}L^{3a+2b+d}T^{-2a-b-d}$
		Reasoning: We substitute the dimensional formulas and compare the powers of M, L, and T.
	4. **Equate the powers of M, L, and T on both sides:**
		* For M: 0 = -a + b  =>  a = b
		* For L: 1 = 3a + 2b + d
		* For T: 0 = -2a - b - d
		Reasoning: We obtain a system of equations.
	5. **Solve the system of equations:**
		From a = b, substitute in the other equations:
		1 = 3a + 2a + d  => 1 = 5a + d
		0 = -2a - a - d  => 0 = -3a - d  => d = -3a
		Substitute d = -3a into 1 = 5a + d:
		1 = 5a - 3a => 1 = 2a => a = 1/2
		Since a = b, b = 1/2
		d = -3a = -3/2
		Reasoning: Solving the system of equations to find the values of a, b, and d.
	6. **Substitute the values of a, b, and d back into the equation for k:**
		$k = G^{1/2} h^{1/2} c^{-3/2}$
		Reasoning: We plug the derived exponents into the initial equation.
	7. **Final Result:**
		$k = {h^{1/2}}{c^{ - 3/2}}{G^{1/2}}$
		Verification: The result matches option A.
 ## OPTION ANALYSIS
[A] ✓: This option correctly gives the dimensions of the radius of gyration when G, h and c are fundamental units.
[B] :x: This option has the wrong power for the speed of light.
[C] :x: This option has the wrong power for the gravitational constant.
[D] :x: This option has the wrong power for the Planck constant.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. **Incorrect dimensional formulas**: Using wrong dimensional formulas for G, h, or c will lead to incorrect results. Always double-check these formulas.
		2. **Errors in solving equations**: Mistakes in solving the system of equations can give incorrect powers. Ensure that each step is carried out carefully.
	**Quick Tips**:
		1. **Remember the dimensions of common quantities**: Knowing the dimensions of basic quantities like force, energy, and power can help in quickly verifying the dimensional formulas of more complex quantities.
		2. **Practice similar problems**: Practice more problems using dimensional analysis to get a better grasp of the method.

P.NARASIMHA · Answer

The correct Option is "${h^{1/2}}{c^{ - 3/2}}{G^{1/2}}$"

P.NARASIMHA · Answer

${h^{1/2}}{c^{3/2}}{G^{1/2}}$

P.NARASIMHA · Answer

${h^{1/2}}{c^{ - 3/2}}{G^{ - 1/2}}$

P.NARASIMHA · Answer

${h^{ - 1/2}}{c^{ - 3/2}}{G^{1/2}}$

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