Calculate the radius of gyration for a plane figure

[shishykish]

Homework Statement

A plane figure is bounded by the curve xy = 4, the x-axis and the ordinates at x = 2 and x =4. Calculate

Relevant Equations

$$I = Ak^2$$
$$xy=4$$
Limits of x= 2 to x=4

Area of strip = $$/delta x /times y$$

Moment of area for strip $$= Area /times (radius of gyration)^2 $$
$$ = /delta x /times y /times (/frac{y^2}{3})^2 $$

Why is the radius of gyration squared term $$y^2$ over 3? $$

 

[haruspex]

Homework Statement: A plane figure is bounded by the curve xy = 4, the x-axis and the ordinates at x = 2 and x =4. Calculate
Relevant Equations: $$I = Ak^2$$
$$xy=4$$
Limits of x= 2 to x=4

Area of strip = $$\delta x \times y$$

Moment of area for strip $$= Area \times (radius of gyration)^2 $$
$$ = \delta x \times y \times (\frac{y^2}{3})^2 $$

I have corrected your LaTeX symbols from "/" to "\".
Presumably you meant $\delta x \times y \times (\frac{y^2}{3})$

Why is the radius of gyration squared term $$y^2/ 3$$?

What is the radius of gyration of a uniform rod?

 

[shishykish]

I have corrected your LaTeX symbols from "/" to "\".
Presumably you meant $\delta x \times y \times (\frac{y^2}{3})$
[]
What is the radius of gyration of a uniform rod?

radius of gyration for a uniform rod is $$ k= \sqrt{ \frac{a^3}{3} }$$from what I remember of deriving it from my notes. So we are actually taking this shape as an elementary strip between the limits of 2 to 4. The total moment of area will be the $$ I_{area} = area \times (radius of gyration)^2 $$

 

[Steve4Physics]

Homework Statement: A plane figure is bounded by the curve xy = 4, the x-axis and the ordinates at x = 2 and x =4. Calculate

Presumably the missing phrase at the end of the sentence is something like: "the radius of gyration about the x-axis".

It is important that the required axis of rotation is clear. For example it could easily be the axis perpendicular to the plane passing through the centre of mass.

 

[shishykish]

Ha! Yes it is. I really could learn how to copy and paste properly!

 

[shishykish]

I have corrected your LaTeX symbols from "/" to "\".
Presumably you meant $\delta x \times y \times (\frac{y^2}{3})$

What is the radius of gyration of a uniform rod?

Hi! So I should be thinking conceptually in terms of calculus. Taking an elementary shape then summing over the whole object is one of the most basic approaches.

 

[haruspex]

radius of gyration for a uniform rod is $$ k= \sqrt{ \frac{a^3}{3} }$$

That’s dimensionality wrong. Is it a typo?

So I should be thinking conceptually in terms of calculus.

Yes, but it is ok to use standard results too. In this problem you have a nonstandard 2D shape that can be represented as a collection of 1D standard shapes - thin rectangular strips that can be taken to be so thin as to be 1D rods.
Work from first principles (the radius of gyration of a point mass at a given distance from an axis) and integrate twice, or skip the first integration by quoting the result for a 1D rod.