The correct order in which the dimensions of time decreases in the following physical quantities
A) Power
B) Modulus of elasticity
C) Moment of inertia
D) Angular momentum

Options

A. A,B,C,D
B. C,D,A, B
C. A,C,D,B
D. C,D,B,A

Answer

To solve the problem of determining the correct order in which the dimensions of time decrease for the given physical quantities, we will analyze each quantity step by step.

### Step 1: Determine the dimensions of each physical quantity.

1. **Power (P)**:
- Power is defined as work done per unit time.
- The dimension of work (W) is given by:
\[ [W] = [F] \cdot [d] = [m \cdot a] \cdot [d] = [m \cdot (l \cdot t^{-2})] \cdot [l] = [m \cdot l^2 \cdot t^{-2}] \]
- Therefore, the dimension of power is:
\[ [P] = \frac{[W]}{[t]} = \frac{[m \cdot l^2 \cdot t^{-2}]}{[t]} = [m \cdot l^2 \cdot t^{-3}] \]

2. **Modulus of Elasticity (E)**:
- Modulus of elasticity is defined as stress divided by strain.
- Strain is dimensionless, and stress is defined as force per unit area.
- The dimension of stress is:
\[ [\text{Stress}] = \frac{[F]}{[A]} = \frac{[m \cdot l \cdot t^{-2}]}{[l^2]} = [m \cdot l^{-1} \cdot t^{-2}] \]
- Thus, the dimension of modulus of elasticity is:
\[ [E] = [\text{Stress}] = [m \cdot l^{-1} \cdot t^{-2}] \]

3. **Moment of Inertia (I)**:
- Moment of inertia is defined as mass times the square of the distance from the axis of rotation.
- Its dimension is:
\[ [I] = [m] \cdot [r^2] = [m] \cdot [l^2] = [m \cdot l^2] \]

4. **Angular Momentum (L)**:
- Angular momentum is defined as the product of mass, velocity, and radius.
- The dimension of angular momentum is:
\[ [L] = [m] \cdot [v] \cdot [r] = [m] \cdot [l \cdot t^{-1}] \cdot [l] = [m \cdot l^2 \cdot t^{-1}] \]

### Step 2: Compile the dimensions of time for each quantity.

- **Power**: \( [P] = [m \cdot l^2 \cdot t^{-3}] \) → Time dimension: \( t^{-3} \)
- **Modulus of Elasticity**: \( [E] = [m \cdot l^{-1} \cdot t^{-2}] \) → Time dimension: \( t^{-2} \)
- **Moment of Inertia**: \( [I] = [m \cdot l^2] \) → Time dimension: \( t^{0} \)
- **Angular Momentum**: \( [L] = [m \cdot l^2 \cdot t^{-1}] \) → Time dimension: \( t^{-1} \)

### Step 3: Arrange the quantities in order of decreasing time dimensions.

Now we can arrange the quantities based on the power of time in their dimensions:

1. **Moment of Inertia (t^0)**: No time dimension
2. **Angular Momentum (t^{-1})**: One negative power of time
3. **Modulus of Elasticity (t^{-2})**: Two negative powers of time
4. **Power (t^{-3})**: Three negative powers of time

### Final Order:
- Moment of Inertia (I) > Angular Momentum (L) > Modulus of Elasticity (E) > Power (P)

Thus, the correct order in which the dimensions of time decrease is:
**C (Moment of Inertia) > D (Angular Momentum) > B (Modulus of Elasticity) > A (Power)**.