Definitions [15]
Answer in one sentence.
Define elasticity.
If a body regains its original shape and size after removal of the deforming force, it is called an elastic body and the property is called elasticity.
A body that regains its original shape and size completely and instantaneously upon removal of the deforming force is said to be perfectly elastic.
A body that regains its original shape and size after removal of the deforming force is called an elastic body, and the property is called elasticity.
A body that does not regain its original shape and size and retains its altered shape or size upon removal of the deforming force is called a plastic body, and the property is called plasticity.
Answer in one sentence.
Define strain.
The strain is defined as the ratio of change in dimensions of the body to its original dimensions.
Strain = `"change in dimensions"/"original dimensions"`
The internal restoring force per unit area of a body is called stress.
Strain is defined as the ratio of the change in dimensions of the body to its original dimensions.
The modulus of elasticity of a material is the ratio of stress to the corresponding strain. It is defined as the slope of the stress-strain curve in the elastic deforming region and depends on the nature of the material.
\[\frac {stress}{strain}\] = Constant
The constant is called the modulus of elasticity.
A graph drawn by taking tensile strain along the x-axis and tensile stress along the y-axis, obtained by gradually increasing the load on a metal wire suspended vertically from a rigid support until the wire breaks, and measuring the elongation produced during each step.
Answer in one sentence.
What do you mean by elastic hysteresis?
In case of some materials like vulcanized rubber, when the stress applied on a body decreases to zero, the strain does not return to zero immediately. The strain lags behind the stress. This lagging of strain behind the stress is called elastic hysteresis.
"Young’s modulus is the ratio of longitudinal stress to longitudinal strain."
"Young’s modulus is the ratio of longitudinal stress to longitudinal strain."
"Bulk modulus is defined as the ratio of volume stress to volume strain."
"Shear modulus or modulus of rigidity: It is defined as the ratio of shear stress to shear strain within elastic limits."
Within elastic limit, the ratio of lateral strain to the linear strain is called the Poisson's ratio.
Formulae [7]
- \[|\vec{F}|\] is the magnitude of the internal restoring force (which is equal to the external applied deforming force).
- A is the area over which the force acts.
- SI unit of stress: N m-2 or pascal (Pa).
- Dimensions of stress: [L-1M1T-2].
The mathematical expression for Young's modulus (Y) is:
Y = \[\frac{MgL}{\pi r^2l}\]
Where:
- Y = Young’s Modulus
- M = Mass of the load attached
- g = Acceleration due to gravity
- L = Original length of the wire
- r = Radius of the wire cross-section
- l = Extension or elongation produced in the wire
The mathematical expression for Young's modulus (Y) is:
Y = \[\frac{MgL}{\pi r^2l}\]
Where:
- Y = Young’s Modulus
- M = Mass of the load attached
- g = Acceleration due to gravity
- L = Original length of the wire
- r = Radius of the wire cross-section
- l = Extension or elongation produced in the wire
The mathematical representation of Bulk Modulus (K) is:
K = \[\frac{\text{Volume Stress}}{\text{Volume Strain}}\]
K = \[\frac{dP}{\left(\frac{dV}{V}\right)}\] = V \[\frac {dP}{dV}\]
Where:
- K: Bulk Modulus
- dP: Change in pressure (Volume Stress)
- dV: Change in volume
- V: Original volume
The formula for modulus of rigidity is:
η = \[\frac{\text{Shear Stress}}{\text{Shear Strain}}=\frac{F/A}{\theta}=\frac{F}{A\cdot\theta}\]
Where:
- η = Modulus of rigidity (Pa or N/m²)
- F = Tangential force applied (N)
- A = Cross-sectional area on which force acts (m²)
- θ = Shear strain = Δl/l (in radians)
- Δl = Displacement of the upper surface relative to the lower surface (m)
- l = Original height of the block (m)
SI Unit: Pascal (Pa) or N/m²
Dimensional Formula: M¹L⁻¹T⁻²
\[\sigma=\frac{\text{Lateral strain}}{\text{Linear strain}}=\frac{\frac{d}{D}}{\frac{\Delta l}{l}}=\frac{d\cdot l}{D\cdot\Delta l}\]
Where:
- σ = Poisson's ratio
- l = original length of the wire
- ∆l = increase or decrease in length of the wire
- D = original diameter of the wire
- d = corresponding change in diameter of the wire
Important Note: Poisson's ratio has no unit. It is dimensionless.
