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What is meant by elastic potential energy? Derive an expression for the elastic potential energy of stretched wire. Prove that its elastic energy density is equal to 1/2 stress x strain.
Definition of Elastic Potential Energy: Elastic potential energy is the energy stored in an elastic material when it is deformed, such as when it is stretched or compressed. This energy is released when the material returns to its original shape. Derivation of Elastic Potential Energy: 1. Consider aRead more
Definition of Elastic Potential Energy:
Elastic potential energy is the energy stored in an elastic material when it is deformed, such as when it is stretched or compressed. This energy is released when the material returns to its original shape.
Derivation of Elastic Potential Energy:
1. Consider a wire of length L and cross-sectional area A with Young’s modulus Y.
2. The tensile force F stretches the wire by an amount x.
3. Stress in the wire is given by
Stress (σ) = F / A
4. Strain in the wire is given by
Strain (ε) = x / L
5. Young’s modulus is defined as
Y = σ / ε = (F/A) / (x/L) => F = (Y * A * x) / L
6. The work done (W) in stretching the wire is given by the area under the stress-strain curve:
W = ∫(from 0 to x) F dx = ∫(from 0 to x) (Y * A * (x/L)) dx
7. Integrating:
W = (Y * A / L) * ∫(from 0 to x) x dx
= (Y * A / L) * [x²/2] (from 0 to x) = (Y * A / L) * (x²/2)
8. Hence, the elastic potential energy (U) stored in the wire is:
U = (Y * A * x²) / (2L)
Proof of Elastic Energy Density is 1/2 Stress x Strain:
1. The density of elastic energy:
is energy per volume
u = U / (A * L)
= [(Y * A * x²) / (2L)] / (A * L)
= (Y * x²) / (2L²)
2. If one starts by assuming the presence of Young’s modulus:
Y = (F / A) / (x / L)
3. Thus, the stress is:
Stress (σ) = F / A = (Y * x) / L
4. And strain is:
Strain (ε) = x / L
5. Substituting for stress and strain:
u = (1/2) * σ * ε
Hence, the elastic energy density is equal to 1/2 stress x strain.
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What is elastic fatigue? What is its importance?
Elastic fatigue is a condition in which a material loses its elastic properties slowly over time as it experiences repeated or cyclic loading and unloading. Continuous cycles of stress cause the material to lose its elastic recovery properties, thus failing to return to its original shape. Elastic fRead more
Elastic fatigue is a condition in which a material loses its elastic properties slowly over time as it experiences repeated or cyclic loading and unloading. Continuous cycles of stress cause the material to lose its elastic recovery properties, thus failing to return to its original shape. Elastic fatigue is very important in engineering and in material selection, especially when it comes to components in bridges, vehicles, and machinery that are subject to fluctuating loads.
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Define Poisson’s ratio.
Poisson's ratio (ν) is defined as the ratio of the transverse strain to the axial strain when a material is subjected to uniaxial stress. It denotes how much a material deforms in the lateral direction when it is stretched or compressed along its length. Mathematically it can be represented as folloRead more
Poisson’s ratio (ν) is defined as the ratio of the transverse strain to the axial strain when a material is subjected to uniaxial stress. It denotes how much a material deforms in the lateral direction when it is stretched or compressed along its length. Mathematically it can be represented as follows:
u = – (transverse strain)/(axial strain) = – (Δd/d)/(ΔL/L)
Where:
– Δd is the change in diameter (transverse deformation),
– d is the original diameter,
– ΔL is the change in length (axial deformation),
– L is the original length.
Important Points
Poisson’s ratio is a dimensionless quantity.
It generally lies between 0 and 0.5 for most materials with values near to 0.5 showing that the material is almost incompressible.
A value of 0 means that there is no transverse deformation when the material is stretched or compressed.
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A given quantity of an ideal gas is at pressure P and absolute temperature T. The isothermal bulk modulus of the gas is
To find the isothermal bulk modulus K of an ideal gas we can use the formula: K = - V (∂P/∂V)_T For an ideal gas at constant temperature (isothermal) the relation between pressure P and volume V is given by Boyle's Law: PV = nRT Differentiating this equation while keeping the temperature constant giRead more
To find the isothermal bulk modulus K of an ideal gas we can use the formula:
K = – V (∂P/∂V)_T
For an ideal gas at constant temperature (isothermal) the relation between pressure P and volume V is given by Boyle’s Law:
PV = nRT
Differentiating this equation while keeping the temperature constant gives us:
∂P/∂V = -nRT/V²
Thus the isothermal bulk modulus becomes:
K = -V (-nRT/V²) = nRT/V
Since nRT = PV we can substitute that in the equation too;
K = PV/V = P
Thus the isothermal bulk modulus of the gas is:
K = P
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A wire of length L, and cross-sectional area A is made of a material of Young’s modulus Y. If the wire is stretched by an amount x, the work done is
To calculate the work done W by stretching a wire of length L and cross-sectional area A through an amount x, we could use the following relationship between stress strain and Young's modulus: The stress in the wire, σ, can be found by the relation as follows: σ = F/A where F represents the appliedRead more
To calculate the work done W by stretching a wire of length L and cross-sectional area A through an amount x, we could use the following relationship between stress strain and Young’s modulus:
The stress in the wire, σ, can be found by the relation as follows:
σ = F/A
where F represents the applied force. The strain ε is described by:
ε = x/L
As related by Young’s modulus Y,
Y = σ/ε = (F/A)/(x/L)
From this we can write the force F:
F = (YAx)/L
The work done W when the wire is stretched by an amount x is given by the area under the stress-strain curve which is the integral of force over displacement:
W = ∫ F dx = ∫ (YAx/L) dx
Evaluating the integral we get:
W = (Y A/L) ∫ x dx = (Y A/L) * [x²/2] from 0 to x = (Y A/L) * (x²/2)
Work done is hence,
W = (Y A x²)/(2 L)
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