Dimensional Analysis

Dimensions of gravitational potential and dimension of latent heat

Mobility,

- Magnetic induction (B) is measured in Tesla (T), and its dimensions are . - Electric field intensity (E) is measured in Volts per meter (V/m), and its dimensions are . Now, calculating the dimensions of : Thus, the correct answer is (D) .

In dimensional analysis, the dimensions of physical quantities are represented in terms of the fundamental quantities such as mass , length , and time . - Power has the dimension , which is the rate at which work is done. - Work and Energy both have the dimension , since they are both forms of energy. - Momentum has the dimension . The given dimension corresponds to the dimension of power since power is the rate of energy transfer, and its dimension is . Thus, the correct answer is Power.

Concept:

Step 1: Dimensional analysis.

Step 1: Understanding the Concept:
Moment of inertia has dimensions . Angular velocity has dimensions .

Step 2: Detailed Explanation:
has dimensions:
This is the dimensional formula for rotational kinetic energy.

Step 3: Final Answer:
.

Concept:
• Inductive reactance
• Capacitive reactance has same dimension as resistance
• Angular frequency

Step 1: Dimensions


Step 2: Compute
Final Conclusion:

Step 1: Understanding the Concept:
The principle of dimensional homogeneity states that an equation is dimensionally correct if the dimensions of all terms on both sides are the same. We can use this to find the unknown dimensions of a physical quantity in a given formula.

Step 2: Key Formula or Approach:
Write the dimensional formula for each known variable in the equation .
Isolate the unknown quantity and solve for its dimensions. Compare the result with the dimensions of the given options.

Step 3: Detailed Explanation:
The dimensions of the given quantities are:
- Time period ( ):
- Constant ( ): Dimensionless, so
- Density ( ):
- Radius ( ): , so
Substitute these into the given formula:


Square both sides to remove the square root:

Rearrange to solve for the dimensions of :

Now, let's find the dimensions of the given options:
(A) Surface tension ( ): . This matches.
(B) Restoring force ( ): . Does not match.
(C) Coefficient of viscosity ( ): . Does not match.

Step 4: Final Answer:
The dimension of is the same as surface tension.

Concept:
• Check correct SI units

Step 1: Verify units
(A) Compressibility \checkmark
(B) Latent heat = \checkmark
(C) Intensity = \checkmark
(D) Energy density should be
(E) Impulse = \checkmark Final Conclusion:
Option (D) is incorrect unit.

Step 1: Understanding the Concept
Dimensional analysis involves expressing physical quantities in terms of fundamental dimensions: Mass ( ), Length ( ), and Time ( ). We derive the dimensions of from Newton's Law of Universal Gravitation.

Step 2: Key Formula or Approach
1. Newton's Law: .
2. Rearranging for : .

Step 3: Detailed Explanation
1. Dimensions of Force ( ) = .
2. Dimensions of distance squared ( ) = .
3. Dimensions of mass product ( ) = .
4. Substitute into the formula for :
5. Simplify the expression:

Step 4: Final Answer
The dimensions of the gravitational constant are .