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A bar of circular cross-section varies uniformly from a cross-section 2D to D. If extension of the bar is calculated treating it as a bar of average diameter, then the percentage error will be
- a)10
- b)25
- c)33
- d)50
Correct answer is option 'A'. Can you explain this answer?
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A bar of circular cross-section varies uniformly from a cross-section ...
Actually,
The average diameter of bar
Approximate extension
% Error
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A bar of circular cross-section varies uniformly from a cross-section ...
Problem Statement:
A bar of circular cross-section varies uniformly from a cross-section 2D to D. If extension of the bar is calculated treating it as a bar of average diameter, then the percentage error will be
Solution:
Given, the bar of circular cross-section varies uniformly from a cross-section 2D to D. Let the length of the bar be L and the Young's modulus of elasticity be E.
We need to calculate the percentage error in the extension of the bar when it is treated as a bar of average diameter.
Calculating the extension of the bar using the actual diameter:
The extension of the bar can be calculated using the formula:
ΔL = FL/EA
where F is the applied force, A is the area of cross-section, and E is the Young's modulus of elasticity.
The area of cross-section varies from 4πD^2 to πD^2/4. So, the average area of cross-section can be calculated as:
(A1 + A2)/2 = (4πD^2 + πD^2/4)/2 = 9πD^2/8
Therefore, the extension of the bar can be calculated as:
ΔL_actual = F*L/(E*(9πD^2/8))
Calculating the extension of the bar using the average diameter:
If we treat the bar as a bar of average diameter, then the area of cross-section is given by:
A_avg = πD_avg^2/4
where D_avg is the average diameter of the bar, given by:
D_avg = (2D + D)/2 = 3D/2
Therefore, the extension of the bar can be calculated as:
ΔL_avg = F*L/(E*(π(3D/2)^2/4))
Calculating the percentage error:
The percentage error in the extension of the bar can be calculated as:
% error = |(ΔL_actual - ΔL_avg)/ΔL_actual| * 100
Substituting the values of ΔL_actual and ΔL_avg, we get:
% error = |(9πD^2/8)/(π(3D/2)^2/4) - 1| * 100
Simplifying, we get:
% error = |9/16 - 1| * 100 = 10%
Therefore, the percentage error in the extension of the bar when it is treated as a bar of average diameter is 10%. Hence, the correct option is (a) 10.
A bar of circular cross-section varies uniformly from a cross-section 2D to D. If extension of the bar is calculated treating it as a bar of average diameter, then the percentage error will be
Solution:
Given, the bar of circular cross-section varies uniformly from a cross-section 2D to D. Let the length of the bar be L and the Young's modulus of elasticity be E.
We need to calculate the percentage error in the extension of the bar when it is treated as a bar of average diameter.
Calculating the extension of the bar using the actual diameter:
The extension of the bar can be calculated using the formula:
ΔL = FL/EA
where F is the applied force, A is the area of cross-section, and E is the Young's modulus of elasticity.
The area of cross-section varies from 4πD^2 to πD^2/4. So, the average area of cross-section can be calculated as:
(A1 + A2)/2 = (4πD^2 + πD^2/4)/2 = 9πD^2/8
Therefore, the extension of the bar can be calculated as:
ΔL_actual = F*L/(E*(9πD^2/8))
Calculating the extension of the bar using the average diameter:
If we treat the bar as a bar of average diameter, then the area of cross-section is given by:
A_avg = πD_avg^2/4
where D_avg is the average diameter of the bar, given by:
D_avg = (2D + D)/2 = 3D/2
Therefore, the extension of the bar can be calculated as:
ΔL_avg = F*L/(E*(π(3D/2)^2/4))
Calculating the percentage error:
The percentage error in the extension of the bar can be calculated as:
% error = |(ΔL_actual - ΔL_avg)/ΔL_actual| * 100
Substituting the values of ΔL_actual and ΔL_avg, we get:
% error = |(9πD^2/8)/(π(3D/2)^2/4) - 1| * 100
Simplifying, we get:
% error = |9/16 - 1| * 100 = 10%
Therefore, the percentage error in the extension of the bar when it is treated as a bar of average diameter is 10%. Hence, the correct option is (a) 10.
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A bar of circular cross-section varies uniformly from a cross-section 2D to D. If extension of the bar is calculated treating it as a bar of average diameter, then the percentage error will bea)10b)25c)33d)50Correct answer is option 'A'. Can you explain this answer?
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