Which of the following physical quantities have into same dimensions ?

Options

A. momentum and impulse
B. pressure and Young's modulus
C. energy and angular momentum
D. force constant and moment of inertia

Answer

To determine which of the given physical quantities have the same dimensions, we will analyze the dimensions of each quantity step by step.

### Step 1: Calculate the dimension of Momentum
- **Momentum (p)** is defined as the product of mass (m) and velocity (v).
- The dimension of mass (m) is \( [M] \).
- The dimension of velocity (v) is \( [L][T^{-1}] \).

Thus, the dimension of momentum is:
\[ \text{Dimension of momentum} = [M][L][T^{-1}] = [MLT^{-1}] \]

### Step 2: Calculate the dimension of Impulse
- **Impulse (J)** is defined as the product of force (F) and time (t).
- The dimension of force (F) is \( [M][L][T^{-2}] \).
- The dimension of time (t) is \( [T] \).

Thus, the dimension of impulse is:
\[ \text{Dimension of impulse} = [M][L][T^{-2}][T] = [MLT^{-1}] \]

### Step 3: Compare Momentum and Impulse
From Steps 1 and 2, we see that:
\[ \text{Dimension of momentum} = [MLT^{-1}] \]
\[ \text{Dimension of impulse} = [MLT^{-1}] \]
**Conclusion:** Momentum and impulse have the same dimensions.

### Step 4: Calculate the dimension of Pressure
- **Pressure (P)** is defined as force (F) per unit area (A).

Thus, the dimension of pressure is:
\[ \text{Dimension of pressure} = \frac{[M][L][T^{-2}]}{[L^2]} = [ML^{-1}T^{-2}] \]

### Step 5: Calculate the dimension of Young's Modulus
- **Young's Modulus (Y)** is defined as stress (force per unit area) divided by strain (change in length/original length).

Thus, the dimension of Young's modulus is:
\[ \text{Dimension of Young's Modulus} = \frac{[M][L][T^{-2}]}{[L^2]} = [ML^{-1}T^{-2}] \]

### Step 6: Compare Pressure and Young's Modulus
From Steps 4 and 5, we see that:
\[ \text{Dimension of pressure} = [ML^{-1}T^{-2}] \]
\[ \text{Dimension of Young's Modulus} = [ML^{-1}T^{-2}] \]
**Conclusion:** Pressure and Young's modulus have the same dimensions.

### Step 7: Calculate the dimension of Energy
- **Energy (E)** is defined as work done, which is force (F) times distance (d).

Thus, the dimension of energy is:
\[ \text{Dimension of energy} = [M][L][T^{-2}][L] = [ML^2T^{-2}] \]

### Step 8: Calculate the dimension of Angular Momentum
- **Angular Momentum (L)** is defined as the product of mass (m), velocity (v), and radius (r).

Thus, the dimension of angular momentum is:
\[ \text{Dimension of angular momentum} = [M][L][T^{-1}][L] = [ML^2T^{-1}] \]

### Step 9: Compare Energy and Angular Momentum
From Steps 7 and 8, we see that:
\[ \text{Dimension of energy} = [ML^2T^{-2}] \]
\[ \text{Dimension of angular momentum} = [ML^2T^{-1}] \]
**Conclusion:** Energy and angular momentum do not have the same dimensions.

### Step 10: Calculate the dimension of Force Constant
- **Force Constant (k)** is defined as force (F) per unit displacement (x).

Thus, the dimension of force constant is:
\[ \text{Dimension of force constant} = \frac{[M][L][T^{-2}]}{[L]} = [MT^{-2}] \]

### Step 11: Calculate the dimension of Moment of Inertia
- **Moment of Inertia (I)** is defined as mass (m) times the square of the distance (r).

Thus, the dimension of moment of inertia is:
\[ \text{Dimension of moment of inertia} = [M][L^2] = [ML^2] \]

### Step 12: Compare Force Constant and Moment of Inertia
From Steps 10 and 11, we see that:
\[ \text{Dimension of force constant} = [MT^{-2}] \]
\[ \text{Dimension of moment of inertia} = [ML^2] \]
**Conclusion:** Force constant and moment of inertia do not have the same dimensions.

### Final Conclusion
The physical quantities that have the same dimensions are:
1. Momentum and Impulse
2. Pressure and Young's Modulus