Which of the following quantities has not been expressed in proper unit?

Options

A. `"torque ": " newton metre"`
B. `"stress ":" newton metre"^(-2)`
C. `"modulus of elasticity ":" newton metre"^(-2)`
D. `"surface tension ":" newton metre"^(-2)`

Answer

To determine which of the given quantities has not been expressed in proper units, we will analyze each option step by step.

### Step 1: Analyze Torque
- Torque (τ) is defined as the product of force (F) and distance (r).
- The formula for torque is:
\[ τ = F \times r \]
- The unit of force (F) is Newton (N) and the unit of distance (r) is meter (m).
- Therefore, the unit of torque is:
\[ \text{Unit of Torque} = \text{Newton} \times \text{meter} = \text{Newton meter} \]
- This means torque is expressed in proper units.

### Step 2: Analyze Modulus of Elasticity
- The modulus of elasticity (Y) is defined as the ratio of stress to strain.
- The formula for modulus of elasticity is:
\[ Y = \frac{F \cdot L}{A \cdot \Delta L} \]
where:
- F = force (in Newtons)
- L = original length (in meters)
- A = cross-sectional area (in square meters)
- ΔL = change in length (in meters)
- The units can be broken down as follows:
- Force (F) = Newton (N)
- Length (L) = meter (m)
- Area (A) = meter² (m²)
- Change in length (ΔL) = meter (m)
- Therefore, the unit of modulus of elasticity is:
\[ Y = \frac{N \cdot m}{m^2 \cdot m} = \frac{N}{m^2} \]
- This means modulus of elasticity is also expressed in proper units.

### Step 3: Analyze Surface Tension
- Surface tension (T) is defined as the force (F) per unit length (L).
- The formula for surface tension is:
\[ T = \frac{F}{L} \]
- The units can be broken down as follows:
- Force (F) = Newton (N)
- Length (L) = meter (m)
- Therefore, the unit of surface tension is:
\[ T = \frac{N}{m} \]
- This means surface tension is also expressed in proper units.

### Conclusion
After analyzing all three quantities:
- Torque is expressed in proper units (Newton meter).
- Modulus of elasticity is expressed in proper units (Newton per meter²).
- Surface tension is expressed in proper units (Newton per meter).

Since all options are expressed in proper units, the question may have been misleading or incorrectly framed. However, based on the analysis, none of the quantities listed are improperly expressed.