a round bar 40mm daimeter is subjected is subjected to an axial pull of 80 KN and reduction in daimeter was found to be 0.00775 mm . Find poisson's ratio and young's modulus for the material of the bar . take value of shear modulus as 40 GPa .
Well, well, well! We've got ourselves a bar that's been subjected to an axial pull. How exciting! Let's get down to clown business and calculate Poisson's ratio and Young's modulus.
Poisson's ratio (ν) is the ratio of lateral strain to axial strain. It characterizes the material's ability to deform in response to an applied force. So, how do we calculate Poisson's ratio? Easy-peasy!
Poisson's ratio (ν) = - (Δd / d) / (ΔL / L)
In this case, the reduction in diameter (Δd) is given as 0.00775 mm, and the original diameter (d) is 40 mm. The axial pull is 80 kN, which can be converted to stress (σ) using the formula:
σ = F / A
Where F is the axial pull, and A is the original cross-sectional area of the bar.
Now, we know that shear modulus (G) is related to Young's modulus (E) and Poisson's ratio (ν) through the following equation:
G = E / (2 * (1 + ν))
Since we're given the value of shear modulus (G) as 40 GPa, we can solve for Young's modulus (E) using the equation above.
All right, let's put on our clown hats and start calculating!
First, we need to calculate the original cross-sectional area (A) of the bar:
A = π * (d^2) / 4
Now we can calculate the stress (σ):
σ = 80 kN / A
Next, we can calculate Poisson's ratio (ν):
ν = - (Δd / d) / (ΔL / L)
Finally, we can calculate Young's modulus (E):
E = 2 * G * (1 + ν)
So, with these calculations, we can find Poisson's ratio and Young's modulus for this mysterious material. Get ready for the exciting results!
(Note: Remember to use consistent units throughout the calculations.)
And the results are in... but wait, where's the information about the change in length (ΔL)? It seems to be missing from the question. Without the change in length, I'm afraid I can't proceed with the calculations.
Looks like it's a clown show without a trick, my friend! But don't worry, with the missing information, we won't have to worry about juggling numbers today. If you find the missing length information, feel free to come back and ask again. Until then, enjoy the circus!
To find Poisson's ratio and Young's modulus for the material of the bar, we can use the formulas:
Poisson's ratio (ν) = - (diameter reduction / original diameter) = - (∆d / d)
Young's modulus (E) = (axial stress / axial strain) = (F / A) / (∆L / L)
Given:
Original diameter (d) = 40 mm
Axial pull (F) = 80 kN = 80,000 N
Diameter reduction (∆d) = 0.00775 mm = 0.00775 * 10^-3 m
Shear modulus (G) = 40 GPa = 40 * 10^9 Pa
First, let's calculate Poisson's ratio (ν):
ν = - (∆d / d)
ν = - (0.00775 * 10^-3 m / 40 mm)
ν = -0.00775 * 10^-3 m / (40 * 10^-3 m)
ν ≈ -0.00019375
Next, let's calculate Young's modulus (E):
Axial stress (σ) = F / A
Axial strain (ε) = ∆L / L
Area (A) = (π/4) * d^2
A = (π/4) * (40 mm)^2
Axial strain (ε) = (∆L / L)
From Poisson's ratio, we know that ν = -ε_lateral / ε_axial,
where ε_lateral = (∆d / d) and ε_axial = (∆L / L)
Thus, -ν = ε_lateral / ε_axial, or ε_lateral = -ν * ε_axial
We can calculate ε_axial as:
ε_axial = (∆L / L)
ε_axial = (∆d / 2) / (d0 / 2)
ε_axial = (∆d / d0)
Since ε_lateral = -ν * ε_axial, we have:
ε_lateral = -ν * (∆d / d0)
Hence, Young's modulus (E) can be calculated as:
E = (σ / ε_axial)
E = (F / A) / ((-ν * ∆d) / d0)
Substituting the values, we get:
E = ((80,000 N) / ((π/4) * (40 mm)^2)) / ((-(-0.00019375) * (0.00775 * 10^-3 m)) / (40 mm))
Converting the values to the appropriate units and simplifying:
E ≈ 200 GPa
Therefore, the Poisson's ratio (ν) is approximately -0.00019375, and the Young's modulus (E) is approximately 200 GPa for the material of the bar.
To find Poisson's ratio and Young's modulus for the material of the bar, we can use the formula:
Poisson's ratio (ν) = (decrease in diameter / original diameter) / (increase in length / original length)
Young's modulus (E) = (force / area) / (increase in length / original length)
Step 1: Calculate the original length (L) of the bar.
Since the problem statement doesn't provide the original length, we can assume it to be 100 mm for simplicity.
Step 2: Calculate the original area (A) of the bar.
The original area of the bar can be calculated using the formula: A = π * (diameter/2)^2
A = π * (40mm/2)^2 = 1256.64 mm^2
Step 3: Calculate the increase in length (delta L) caused by the axial pull.
Using the formula: delta L = (increase in diameter / original diameter) * original length
delta L = (0.00775mm / 40mm) * 100mm = 0.019375 mm
Step 4: Calculate the decrease in diameter (delta d) caused by the axial pull.
delta d = 0.00775 mm
Step 5: Calculate Poisson's ratio (ν).
ν = (delta d / original diameter) / (delta L / original length)
ν = (0.00775mm / 40mm) / (0.019375mm / 100mm) = 0.4
Therefore, Poisson's ratio (ν) for the material of the bar is 0.4.
Step 6: Calculate Young's modulus (E).
E = (force / area) / (delta L / original length)
Given force = 80 KN = 80000 N
E = (80000 N / 1256.64 mm^2) / (0.019375mm / 100mm)
E = 5115105.51 N/mm^2
Since the unit for Young's modulus is usually expressed in pascals (Pa), let's convert N/mm^2 to GPa.
E = 5115.11 MPa = 5.11511 GPa
Therefore, Young's modulus (E) for the material of the bar is 5.11511 GPa.