Problem 969 - JEE Physics

Question

Which is the dimensional formula for modulus of rigidity? (A) \(\mathrm{M}_{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\) (B) \(\mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\) (C) \(\mathrm{M}^{1} \mathrm{~L}^{-2} \mathrm{~T}^{-1}\) (D) \(\mathrm{M}^{1} \mathrm{~L}^{-2} \mathrm{~T}^{-2}\)

Step-by-Step Solution

Answer

The dimensional formula for modulus of rigidity is (B) \(\mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\).

Step 1: Define the formula for shear stress

Shear stress is defined as the force applied per unit area. Mathematically, it is given by the formula: \[ \text{Shear stress} = \frac{\text{Force}}{\text{Area}} \] The dimensional formula for force is given by \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\), and the dimensional formula for the area is given by \(\mathrm{L}^{2}\).

Step 2: Find the dimensional formula for shear stress

Using the formula for shear stress, we can substitute the dimensional formulas for force and area: \[ \text{Shear stress} = \frac{\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}}{\mathrm{L}^{2}} \] Now, simplify the expression: \[ \text{Shear stress} = \mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{-2} \]

Step 3: Define the formula for shear modulus

The formula for shear modulus or modulus of rigidity is given by the following equation: \[ \text{Shear modulus} = \frac{\text{Shear stress}}{\text{Shear strain}} \] Shear strain is dimensionless, as it is the ratio of displacement or deformation to the initial length. Therefore, it does not have any dimensions.

Step 4: Find the dimensional formula for modulus of rigidity

Since shear strain is dimensionless, the dimensional formula for shear modulus or modulus of rigidity will be the same as the dimensional formula for shear stress derived in Step 2: \[ \text{Modulus of rigidity} = \mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{-2} \] Comparing the result with the given options, we find that: The dimensional formula for modulus of rigidity is (B) \(\mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\).

Key Concepts

Modulus of Rigidity

The Modulus of Rigidity, also known as the shear modulus, is a measure of a material's ability to resist shear deformation. In simple terms, it quantifies how a material will react to forces that attempt to change its shape.
Imagine a thick rubber eraser. When you try to twist it, the resistance you feel is due to its modulus of rigidity. The higher the modulus, the stiffer the material, meaning it will resist deformation more vigorously.

The modulus of rigidity is represented by the symbol 'G' and defined through the relationship between shear stress and shear strain:

• Formula: \( G = \frac{\text{Shear Stress}}{\text{Shear Strain}} \)
• Since shear strain is dimensionless, the dimensional formula for modulus of rigidity is identical to shear stress, which is \( \mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{-2} \)

Shear Stress

Shear stress is a type of stress that causes layers of a material to slide past each other. This concept is important in understanding how forces act on materials and how they will respond.
When force is applied parallel or tangential to the surface of an object, shear stress is induced. Picture a deck of cards being slightly pushed on top while holding the bottom steady; this visualizes how shear stress works in materials.

Mathematically, shear stress (\( \tau \)) is calculated by dividing the force by the area over which the force is exerted:

• Formula: \( \tau = \frac{\text{Force}}{\text{Area}} \)
• The dimensional formula for shear stress is derived as \( \mathrm{M}^{1} \mathrm{~L}^{-1} \mathrm{~T}^{-2} \), using the dimensions for force \( \mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2} \) and area \( \mathrm{L}^{2} \).

Shear Strain

Shear strain describes the deformation of an object in response to shear stress, specifically how much a material changes in shape without any change in volume. It's an important factor in materials science because it helps engineers understand how a material will behave when subjected to forces.
Shear strain is calculated by the relative displacement of layers divided by the distance between those layers. Imagine taking a square block of jelly and shifting the top surface sideways without altering its base; the angle of inclination formed is a measure of shear strain.

Key points about shear strain:

• Nature: It is a dimensionless quantity, which means it doesn't have units. It's simply a ratio of lengths.
• This dimensionless property is why the modulus of rigidity shares dimensions with shear stress, as specified in the relationship \( G = \frac{\text{Shear Stress}}{\text{Shear Strain}} \).