In elasticity, what is the SI unit of the physical quantity known as Youngs modulus, which measures the ratio of stress to strain in a material?

Introduction / Context:
Youngs modulus is a fundamental mechanical property that measures the stiffness of a material. It is defined as the ratio of tensile or compressive stress to the corresponding strain within the elastic limit. Because stress is force per unit area and strain is dimensionless, the unit of Youngs modulus must be the same as the unit of stress. Understanding this helps connect concepts in mechanics with units used in engineering.

Given Data / Assumptions:

• Youngs modulus E is defined as E = stress / strain.
• Stress is force per unit area, so stress has units of newton per square metre.
• Strain is the ratio of change in length to original length and is a pure number with no unit.
• The SI unit for pressure and stress is the pascal (Pa).

Concept / Approach:
Since Youngs modulus is the ratio of stress to strain and strain has no unit, the unit of Youngs modulus is the same as the unit of stress. In SI units, stress is measured in newton per square metre (N/m^2). This unit is given a special name, pascal (Pa). Therefore, the correct unit for Youngs modulus is the pascal. Newton alone is a unit of force, not of stress. Joule is a unit of energy. Erg is an old CGS unit of energy, not used as an SI unit in this context. Hence, pascal is the correct SI unit.

Step-by-Step Solution:
Step 1: Recall the definition of Youngs modulus E = stress / strain.Step 2: Recognise that stress = force / area, with SI units newton per square metre.Step 3: Note that strain is dimensionless because it is a ratio of two lengths.Step 4: Therefore, the unit of E equals the unit of stress, which is N/m^2.Step 5: Remember that in SI, one pascal is defined as one newton per square metre.Step 6: Conclude that Youngs modulus is measured in pascals.

Verification / Alternative check:
Engineering tables list Youngs modulus for steel as about 2 * 10^11 N/m^2 or 200 gigapascals (200 GPa). The fact that both forms are used interchangeably shows that pascal and N/m^2 are the same unit. No standard reference expresses Youngs modulus in joules or ergs, because it is not energy but stress. This reinforces that pascal is the correct SI unit.

Why Other Options Are Wrong:
Newton is a unit of force, suitable for measuring weight or other forces, but Youngs modulus relates to stress, not pure force. Joule is a unit of work or energy, related to force times distance, not stress. Erg is a CGS unit of energy and is again not used to measure stress or Youngs modulus. None of these match the definition or dimensional form of Youngs modulus.

Common Pitfalls:
Students sometimes confuse units of stress and energy, especially because both involve force and distance in different combinations. Another mistake is assuming that any mechanical quantity may be expressed in newtons alone. To avoid these errors, always analyse the definition of the quantity: because Youngs modulus is stress divided by a dimensionless number, its unit must be the same as stress, which is pascal in SI.

Final Answer:
The SI unit of Youngs modulus is the pascal (Pa).