Which of the two have same dimensions ?

Options

A. Force and strain
B. Force and stress
C. Angular velocity and frequency
D. Energy and strain

Answer

To determine which of the two quantities have the same dimensions, we will analyze the dimensions of each option provided in the question.

### Step-by-Step Solution:

1. **Identify the quantities:**
We need to compare the dimensions of the given quantities. The options typically include physical quantities like force, stress, angular velocity, frequency, etc.

2. **Calculate the dimensions of Force:**
- Force (F) is defined as mass (m) times acceleration (a).
- The dimension of mass is [M], and the dimension of acceleration is [L][T^-2].
- Therefore, the dimension of force is:
\[ [F] = [M][L][T^{-2}] = [M L T^{-2}] \]

3. **Calculate the dimensions of Stress:**
- Stress (σ) is defined as force per unit area.
- The dimension of area (A) is [L^2].
- Therefore, the dimension of stress is:
\[ [σ] = \frac{[F]}{[A]} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

4. **Calculate the dimensions of Angular Velocity (ω):**
- Angular velocity is defined as the change in angle (Δθ) per unit time (Δt).
- The dimension of angle is dimensionless (radians are a ratio of lengths).
- Therefore, the dimension of angular velocity is:
\[ [ω] = \frac{[Δθ]}{[Δt]} = \frac{1}{[T]} = [T^{-1}] \]

5. **Calculate the dimensions of Frequency (f):**
- Frequency is defined as the number of cycles per unit time.
- Therefore, the dimension of frequency is:
\[ [f] = \frac{1}{[T]} = [T^{-1}] \]

6. **Compare the dimensions:**
- From the calculations:
- Dimension of Force: \([M L T^{-2}]\)
- Dimension of Stress: \([M L^{-1} T^{-2}]\)
- Dimension of Angular Velocity: \([T^{-1}]\)
- Dimension of Frequency: \([T^{-1}]\)

7. **Conclusion:**
- The only two quantities that have the same dimensions are Angular Velocity (ω) and Frequency (f), both of which have the dimension \([T^{-1}]\).

### Final Answer:
The quantities that have the same dimensions are Angular Velocity (ω) and Frequency (f).