How to use the moment of inertia of solids calculator
- Pick a shape from the list and enter its mass and main size.
- For a hollow cylinder or a box, enter the second size.
- In More options, shift the axis away from the center and enter a slope height for the rolling speed.
- Read the moment of inertia, the radius of gyration and, for round shapes, the speed after rolling down the slope.
Formula
Solid sphere (2/5) m r^2; thin shell (2/3) m r^2; solid cylinder (1/2) m r^2; hoop m r^2
Hollow cylinder (1/2) m (R^2 + r^2); disc about a diameter (1/4) m r^2; cone (3/10) m r^2
Rod: (1/12) m L^2 about the center, (1/3) m L^2 about an end; box (1/12) m (a^2 + b^2)
Parallel axis: I = I_cm + m d^2; radius of gyration k = sqrt(I / m); rolling v = sqrt(2 g h / (1 + I / (m r^2)))
The moment of inertia measures how hard it is to change a body's spin. Mass farther from the axis counts for more, because distance is squared.
A ball, a pipe, a rod and a shifted disc
A solid sphere of mass 2 kg and radius 0.1 m has I = (2/5) x 2 x 0.01 = 0.008 kg m^2 and a radius of gyration of 0.0632 m. Rolling down a 1 m drop it reaches 3.7429 m/s, against 4.4287 m/s if it slid without friction, with 28.57 percent of the energy going into spin.
A hollow cylinder of 2 kg with radii 0.1 and 0.08 m has I = 0.0164 kg m^2. A 1.5 kg rod 1.2 m long has I = 0.72 kg m^2 about its end. A 4 kg disc of radius 0.25 m has I = 0.125 kg m^2 about its center, and 0.485 kg m^2 about an axis 0.3 m away.
Why rolling objects differ
When an object rolls down a slope, gravity gives up the same energy to every object, but part of it goes into spinning. A hoop puts half into spin, a solid cylinder a third and a solid sphere 2/7. So on the same slope a sphere beats a cylinder, which beats a hoop, whatever their masses and radii. The share of energy in spin is shown on the page.
Cross-section properties used for beams, a different quantity, are in the section properties calculator and the beam deflection calculator. For engines see the horsepower from torque calculator.
Choosing the axis
The formulas for each shape apply to the axis named in the shape's label: the symmetry axis for round solids, the center or end perpendicular to a rod, and for a box the axis through its center perpendicular to the face with sides a and b. For any parallel axis, add the mass times the square of the distance from the center-of-mass axis.
Assumptions and limits
- The body is rigid and has uniform density. Hollow shapes have thin walls unless the page gives two radii.
- Rolling is without slipping, starting from rest, and the rolling radius is the outer radius.
- Mass, length and shift can be entered in several units and are converted. Results are in kg m^2.
- If you shift the axis, the rolling speed still uses the moment of inertia about the center, since rolling needs the center axis.
Frequently asked questions
What is the moment of inertia of a solid sphere?
It is 2/5 of the mass times the radius squared, about any axis through the center. A thin hollow shell has 2/3 m r^2 because all its mass is at the full radius.
What is the parallel axis theorem?
It says the moment of inertia about any axis equals the moment about a parallel axis through the center of mass, plus the mass times the square of the distance between the axes.
What is the radius of gyration?
It is the distance from the axis at which all the mass could be placed to give the same moment of inertia: sqrt(I / m). It is a single length that summarizes how spread out the mass is.
Why does a hoop roll slower than a disc?
More of a hoop's mass sits at the rim, so it has a larger moment of inertia for its mass. More of the energy has to go into spinning, which leaves less for forward motion.