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A seamless cylinder of storage capacity of 0.03mᵌis subjected to an internal pressure of 21MPa. The ultimate strength of material of cylinder is 350N/mm².Determine the length of the cylinder if it is twice the diameter of the cylinder.
- a)540mm
- b)270mm
- c)400mm
- d)350mm
Correct answer is option 'A'. Can you explain this answer?
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A seamless cylinder of storage capacity of 0.03mᵌis subjected to...
Explanation: 0.03=πd²L/4 and L=2d.
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A seamless cylinder of storage capacity of 0.03mᵌis subjected to...
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To determine whether the cylinder can withstand the internal pressure without failure, we need to calculate the maximum stress in the cylinder and compare it to the ultimate strength of the material.
We can use the formula for hoop stress in a cylinder:
σ = Pd/2t
Where:
σ = hoop stress
P = internal pressure
d = diameter of cylinder
t = thickness of cylinder
We are given the storage capacity of the cylinder, so we can use the formula for volume of a cylinder to find the diameter:
V = πr^2h
Where:
V = volume
r = radius
h = height
0.03m^3 = πr^2h
r^2h = 0.03/π
r = (0.03/πh)^0.5
The cylinder is seamless, so the thickness is uniform. Let's assume a thickness of 10mm.
d = 2r + 2t
d = 2(0.03/πh)^0.5 + 20mm
Now we can calculate the hoop stress:
σ = (21 x 10^6 N/m^2) x d/2t
σ = (21 x 10^6 N/m^2) x [(2(0.03/πh)^0.5 + 20mm)/20mm]
Simplifying the equation and converting to N/mm^2:
σ = 0.33(0.03/πh)^0.5 + 350 MPa
The maximum stress in the cylinder is a function of the height, so we need to find the height at which the stress is highest. To do this, we can take the derivative of the stress equation with respect to h and set it equal to zero:
dσ/dh = -0.83π^-0.5(0.03/πh)^1.5 = 0
h = (0.03/π)^0.333
Plugging this value of h back into the stress equation, we get:
σ = 0.33(0.03/π(0.03/π)^0.667)^0.5 + 350 MPa
σ = 594.3 MPa
The ultimate strength of the material is 350 N/mm^2, which is less than the maximum stress of 594.3 MPa. Therefore, the cylinder will fail under the given internal pressure.
To determine whether the cylinder can withstand the internal pressure without failure, we need to calculate the maximum stress in the cylinder and compare it to the ultimate strength of the material.
We can use the formula for hoop stress in a cylinder:
σ = Pd/2t
Where:
σ = hoop stress
P = internal pressure
d = diameter of cylinder
t = thickness of cylinder
We are given the storage capacity of the cylinder, so we can use the formula for volume of a cylinder to find the diameter:
V = πr^2h
Where:
V = volume
r = radius
h = height
0.03m^3 = πr^2h
r^2h = 0.03/π
r = (0.03/πh)^0.5
The cylinder is seamless, so the thickness is uniform. Let's assume a thickness of 10mm.
d = 2r + 2t
d = 2(0.03/πh)^0.5 + 20mm
Now we can calculate the hoop stress:
σ = (21 x 10^6 N/m^2) x d/2t
σ = (21 x 10^6 N/m^2) x [(2(0.03/πh)^0.5 + 20mm)/20mm]
Simplifying the equation and converting to N/mm^2:
σ = 0.33(0.03/πh)^0.5 + 350 MPa
The maximum stress in the cylinder is a function of the height, so we need to find the height at which the stress is highest. To do this, we can take the derivative of the stress equation with respect to h and set it equal to zero:
dσ/dh = -0.83π^-0.5(0.03/πh)^1.5 = 0
h = (0.03/π)^0.333
Plugging this value of h back into the stress equation, we get:
σ = 0.33(0.03/π(0.03/π)^0.667)^0.5 + 350 MPa
σ = 594.3 MPa
The ultimate strength of the material is 350 N/mm^2, which is less than the maximum stress of 594.3 MPa. Therefore, the cylinder will fail under the given internal pressure.
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A seamless cylinder of storage capacity of 0.03mᵌis subjected to an internal pressure of 21MPa. The ultimate strength of material of cylinder is 350N/mm².Determine the length of the cylinder if it is twice the diameter of the cylinder.a)540mmb)270mmc)400mmd)350mmCorrect answer is option 'A'. Can you explain this answer?
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