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The radius of gyration of a rectangular section (depth d, width b) from a centroidal axis parallel to the width is
- a)0.57 d
- b)0.28 d
- c)0.28 b
- d)0.57 b
Correct answer is option 'B'. Can you explain this answer?
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Explanation:
The radius of gyration is a measure of how the mass is distributed around an axis of rotation. It is defined as the square root of the ratio of the moment of inertia to the mass. In the case of a rectangular section, the moment of inertia is given by the formula:
I = (1/12) * b * d^3
where b is the width of the section and d is the depth of the section.
The mass of the section can be calculated by multiplying the density of the material by the volume of the section:
m = ρ * b * d * L
where ρ is the density of the material and L is the length of the section.
The radius of gyration can then be calculated as:
k = sqrt(I / m) = sqrt((1/12) * b * d^3 / (ρ * b * d * L)) = sqrt((1/12) * d^2 / (ρ * L)) = (1/√12) * d * sqrt(1 / (ρ * L)) = (1/√12) * d * sqrt(1 / ρ) * sqrt(1 / L)
Since the radius of gyration is parallel to the width of the section, we can rewrite the equation as:
k = (1/√12) * d * sqrt(1 / ρ) * sqrt(1 / L) = (1/√12) * d * sqrt(1 / ρ)
The given options are:
a) 0.57d
b) 0.28d
c) 0.28b
d) 0.57b
Comparing the given options with the derived equation, we can see that option b) 0.28d is the correct answer.
The radius of gyration is a measure of how the mass is distributed around an axis of rotation. It is defined as the square root of the ratio of the moment of inertia to the mass. In the case of a rectangular section, the moment of inertia is given by the formula:
I = (1/12) * b * d^3
where b is the width of the section and d is the depth of the section.
The mass of the section can be calculated by multiplying the density of the material by the volume of the section:
m = ρ * b * d * L
where ρ is the density of the material and L is the length of the section.
The radius of gyration can then be calculated as:
k = sqrt(I / m) = sqrt((1/12) * b * d^3 / (ρ * b * d * L)) = sqrt((1/12) * d^2 / (ρ * L)) = (1/√12) * d * sqrt(1 / (ρ * L)) = (1/√12) * d * sqrt(1 / ρ) * sqrt(1 / L)
Since the radius of gyration is parallel to the width of the section, we can rewrite the equation as:
k = (1/√12) * d * sqrt(1 / ρ) * sqrt(1 / L) = (1/√12) * d * sqrt(1 / ρ)
The given options are:
a) 0.57d
b) 0.28d
c) 0.28b
d) 0.57b
Comparing the given options with the derived equation, we can see that option b) 0.28d is the correct answer.
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The radius of gyration of a rectangular section (depth d, width b) from a centroidal axis parallel to the width isa)0.57 db)0.28 dc)0.28 bd)0.57 bCorrect answer is option 'B'. Can you explain this answer?
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