# Rotational Inertia

## Problems

### practice

- Four point objects of mass
*m*are located at the corners of a square of side*s*as shown in the figure to the right. Determine the moment of inertia of this system if it is rotated about…- the perpendicular bisector of a side
- a side
- a diagonal
- one corner on an axis perpendicular to the plane containing the masses

- Write something.
- Write something.
- Determine the moment of inertia for each of the following shapes. The rotational axis is the same as the axis of symmetry in all but two cases. Use
*M*for the mass of each object.- ring, hoop, cylindrical shell, thin pipe
- annulus, hollow cylinder, thick pipe
- disk, solid cylinder
- spherical shell
- hollow sphere
- solid sphere
- rod, rectangular plate (perpendicular bisector)
- rod, rectangular plate (axis along edge)
- rectangular plate, solid box (axis perpendicular to face)
- cube (axis perpendicular to face)
- cone (rotated about its central axis)
- cone (rotated about its vertex)

### conceptual

- Spacecraft from the US landed on Mars in 1976 and 1997. By communicating with spacecraft on the surface, NASA scientists were able to determine the orientation of Mars in space. Changes in the orientation between 1976 and 1997 were then used to determine the moment of inertia of Mars. Why would anyone care about the moment of inertia of a planet? What good is it to know this quantity?

### algebraic

- For many applications, it's better to have the moment of inertia written in terms of the density of the material it's made out of than the mass of the finished object. Do this for each of the following shapes…
object mass-moment density-moment disk, solid cylinder *I*=^{1}_{2}*MR*^{2}annulus, hollow cylinder, thick pipe *I*=^{1}_{2}*M*(*R*_{2}^{2}+*R*_{1}^{2})solid sphere *I*=^{2}_{5}*MR*^{2}hollow sphere *I*=^{2}_{5}*M**R*_{2}^{5}−*R*_{1}^{5}*R*_{2}^{3}−*R*_{1}^{3}rectangular plate, solid box (axis perpendicular to face) *I*=^{1}_{12}*M*(*L*^{2}+*W*^{2})cube (axis perpendicular to face) *I*=^{1}_{6}*MS*^{2}cone (rotated about its central axis) *I*=^{3}_{10}*MR*^{2}cone (rotated about its vertex) *I*=^{3}_{5}*M*(^{1}_{4}*R*^{2}+*H*^{2}) - Derive the moment of inertia for each of the following flat geometric shapes when they are rotated about a diameter instead of the axis of symmetry using the perpendicular axis theorem.
*I*=_{z}*I*+_{x}*I*_{y}shape axis of symmetry diameter ring or hoop *I*=^{1}_{2}*MR*^{2}annulus *I*=^{1}_{4}*M*(*R*_{2}^{2}+*R*_{1}^{2})disk *I*=^{1}_{4}*MR*^{2}

### calculus

- Derive the moment of inertia for each of the following flat geometric shapes when they are rotated about a diameter instead of the axis of symmetry using the integral equation.
*I*=⌠

⌡*r*^{2}*dm*shape axis of symmetry diameter ring or hoop *I*=^{1}_{2}*MR*^{2}annulus *I*=^{1}_{4}*M*(*R*_{2}^{2}+*R*_{1}^{2})disk *I*=^{1}_{4}*MR*^{2} - Show that the moment of inertia of a cone rotated about its vertex is given by…

Where*I*=^{3}_{5}*M*(^{1}_{4}*R*^{2}+*H*^{2})*R*is the radius of the base and*H*is the height. - Show that the moment of inertia of a cylindrical shell or thin pipe of radius
*R*and length*L*is given by…

…when it is rotated about a diameter through its center of mass.*I*=^{1}_{2}*M*(*R*^{2}+*L*^{2}) - Show that the moment of inertia of a ring torus (a doughnut) with major radius
*R*(the mean radius of the ring) and minor radius*r*(the radius of a cross section) is given by…

…when it is rotated about its axis like a wheel and by…*I*=*MR*^{2}+^{3}_{4}*Mr*^{2}

…when it is rotated about a diameter. (Assume that*I*=^{1}_{2}*MR*^{2}+^{5}_{8}*Mr*^{2}*R*>*r*.) - Show that the moment of inertia of a thin-walled, hollow cubical box with edges of length
*S*and total mass*M*is given by…

…when it is rotated about an axis through its geometric center and perpendicular to one face.*I*=^{5}_{18}*MS*^{2}

### statistical

- earth.txt

The data in this text file gives the density and gravitational field strength of the Earth at various depths below the surface. Using data analysis software (preferably something that can do numerical integration) determine the moment of inertia of the Earth.