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 pairs have the same dimensions, we will analyze each pair one by one.

### Step 1: Analyze Force and Strain
- **Force**: The formula for force is given by Newton's second law, \( F = ma \), where \( m \) is mass and \( a \) is acceleration.
- The unit of force is Newton (N), which can be expressed as:
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \]
- The dimensions of force are:
\[ [F] = [M^1 L^1 T^{-2}] \]

- **Strain**: Strain is defined as the ratio of change in length to the original length, which is a dimensionless quantity.
- Therefore, strain has no dimensions:
\[ [\text{Strain}] = [1] \]

**Conclusion**: Force and strain do not have the same dimensions.

### Step 2: Analyze Force and Stress
- **Stress**: Stress is defined as force per unit area.
- The unit of stress is Pascal (Pa), which can be expressed as:
\[ 1 \, \text{Pa} = 1 \, \text{N/m}^2 \]
- Thus, the dimensions of stress are:
\[ [\text{Stress}] = \frac{[F]}{[A]} = \frac{[M^1 L^1 T^{-2}]}{[L^2]} = [M^1 L^{-1} T^{-2}] \]

**Conclusion**: Force and stress do not have the same dimensions.

### Step 3: Analyze Angular Velocity and Frequency
- **Angular Velocity**: Angular velocity (\( \omega \)) is defined as the rate of change of angular displacement with respect to time.
- The unit of angular velocity is radians per second (rad/s), which has the dimensions:
\[ [\omega] = [T^{-1}] \]

- **Frequency**: Frequency (\( f \)) is defined as the number of occurrences of a repeating event per unit time.
- The unit of frequency is Hertz (Hz), which is also expressed as:
\[ 1 \, \text{Hz} = 1 \, \text{s}^{-1} \]
- Thus, the dimensions of frequency are:
\[ [f] = [T^{-1}] \]

**Conclusion**: Angular velocity and frequency have the same dimensions.

### Step 4: Analyze Energy and Strain
- **Energy**: The unit of energy is Joule (J), which can be expressed as:
\[ 1 \, \text{J} = 1 \, \text{kg} \cdot \text{m}^2/\text{s}^2 \]
- The dimensions of energy are:
\[ [E] = [M^1 L^2 T^{-2}] \]

- **Strain**: As established earlier, strain is dimensionless:
\[ [\text{Strain}] = [1] \]

**Conclusion**: Energy and strain do not have the same dimensions.

### Final Answer
The only pair that has the same dimensions is **Option C: Angular Velocity and Frequency**.

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