How do I calculate the moment of inertia of a torus with given parameters?

[rock.freak667]

Homework Statement

Find the moment of inertia of a torus if mass is m and density $\rho$ is constant.
The cross-sectional radius is 'a' and the radius is R.

Homework Equations

$$I= \int r^2 dm$$

The Attempt at a Solution

Well I looked up the answer to be

$$I_z= m(R^2 + \frac{3}{4}a^2)$$

But I am not sure how to start. Can someone just point me in the right direction?

 

[tiny-tim]

Find the moment of inertia of a torus if mass is m and density $\rho$ is constant.
The cross-sectional radius is 'a' and the radius is R.

Homework Equations

$$I= \int r^2 dm$$

Hi rock.freak667!

(have a rho: ρ )

Do you mean the moment of inertia about its axis of rotational symmetry? And is R the internal radius, or the average radius?

Hint: divide the torus into cylindrical slices of thickness dr, and integrate between R ± a

 

[rock.freak667]

Yes R is the internal radius.


So if I am considering cylindrical shells of thickness dr.

if I draw it in 2d, it makes a circle such that $x^2+z^2=a^2$


the volume of a cylindrical shell is

$$dV= \pi z^2 dr$$

dV=pi z² dr

so the moment of inertia of the small cylindrical element is

$$dI_c = \frac{1}{2} (\rho \pi z^2 dr) z^2$$

dIc= (1/2) (p*pi*z² dr)z²

But this would give me the inertia not about the z-axis right but the axis perpendicular to the cylindrical shell. Which is not about the z-axis.

and also I would be integrating z w.r.t. r

 

[tiny-tim]

Hi rock.freak667!

the volume of a cylindrical shell is

dV=pi z² dr

I'm not sure what your slices are, but that looks like the volume of a cylinder.

A cylindrical shell is the (thickened) surface of a cylinder.

 

[rock.freak667]

Hi rock.freak667!


I'm not sure what your slices are, but that looks like the volume of a cylinder.

A cylindrical shell is the (thickened) surface of a cylinder.

so dV= (2pi*z)x dr ? Not sure on the surface area of cylindrical shell that I'm considering

 

[tiny-tim]

(have a pi: π )

so dV= (2pi*z)x dr ? Not sure on the surface area of cylindrical shell that I'm considering

(2π*z)x dr ?

what are z and x ?

slice it with a cookie cutter of radius r, then slice it again with a cookie cutter of radius r + dr