Young's modulus of a material has the same units as (A) pressure (B) strain (C) compressibility (D) Force

Options

A. pressure
B. strain
C. compressibility
D. force

Explanation

To determine which of the given options has the same units as Young's modulus, we first need to understand the definition and units of Young's modulus.

### Step 1: Definition of Young's Modulus
Young's modulus (Y) is defined as the ratio of stress to strain.

### Step 2: Understanding Stress and Strain
- **Stress** is defined as the force (F) applied per unit area (A):
\[ \text{Stress} = \frac{F}{A} \]
- **Strain** is defined as the change in length (ΔL) divided by the original length (L):
\[ \text{Strain} = \frac{\Delta L}{L} \]

### Step 3: Units of Stress
The units of stress can be derived as follows:
- Force (F) has units of Newton (N), which is equivalent to \( \text{kg} \cdot \text{m/s}^2 \).
- Area (A) has units of \( \text{m}^2 \).

Thus, the units of stress are:
\[ \text{Units of Stress} = \frac{\text{Force}}{\text{Area}} = \frac{N}{m^2} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}^2} = \frac{\text{kg}}{\text{m} \cdot \text{s}^2} \]

### Step 4: Units of Strain
Strain is a dimensionless quantity since it is a ratio of two lengths:
\[ \text{Units of Strain} = \frac{L}{L} = 1 \quad (\text{dimensionless}) \]

### Step 5: Units of Young's Modulus
Now, substituting the units of stress and strain into the formula for Young's modulus:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{\frac{F}{A}}{\text{Strain}} = \frac{\frac{N}{m^2}}{1} = \frac{N}{m^2} \]
Thus, the units of Young's modulus are:
\[ \text{Units of Young's Modulus} = \frac{\text{kg}}{\text{m} \cdot \text{s}^2} \cdot \frac{1}{\text{m}^2} = \frac{\text{kg}}{\text{m} \cdot \text{s}^2} = \text{Pressure} \]

### Step 6: Comparison with Given Options
Now, we compare the units of Young's modulus with the options given:
- (A) Pressure: Units are \( \frac{\text{kg}}{\text{m} \cdot \text{s}^2} \) (same as Young's modulus)
- (B) Strain: Dimensionless (not the same)
- (C) Compressibility: Different units (not the same)
- (D) Force: Units are \( \text{kg} \cdot \text{m/s}^2 \) (not the same)

### Conclusion
The correct answer is:
**(A) Pressure**