Calculate how a rolling body’s kinetic energy is divided between translation of its centre of mass and rotation about its centre of mass. Enter mass in kilograms, centre-of-mass speed in metres per second and the dimensionless inertia factor β = I/(mR²). The default β = 0.5 represents a uniform solid cylinder or disc rolling on its circular rim.
| Body | β = I/(mR²) | Value to enter |
|---|---|---|
| Uniform solid sphere | 2/5 | 0.4 |
| Uniform solid cylinder or disc, about its symmetry axis | 1/2 | 0.5 |
| Thin uniform spherical shell, about a diameter | 2/3 | 0.6666666667 (approximate) |
| Thin hoop or thin cylindrical shell, about its symmetry axis | 1 | 1 |
A uniform solid cylinder of mass 2 kg rolls without slipping at 3 m/s. With β = 0.5, its translational energy is ½ × 2 × 3² = 9 J. Its rotational energy is 0.5 × 9 = 4.5 J, giving a total of 13.5 J. These are the calculator’s default inputs.
This model assumes a rigid body with a fixed rolling radius and a centre-of-mass rotation axis for which v = ωR. The surface is stationary in the reference frame used to measure speed. It calculates instantaneous kinetic energy; it does not determine whether available friction is sufficient to prevent slipping, or model rolling resistance, deformation or energy losses.
Use positive mass for a physical body and nonnegative speed and inertia factor. Zero mass is accepted only as a mathematical limit; zero speed gives zero kinetic energy. β = 0 is a mathematical zero-inertia limit. Use an inertia factor appropriate to the body’s actual mass distribution and rolling radius; the table applies only to the stated ideal shapes.
Unavailable results indicate a calculation or numeric-range problem and must not be interpreted as zero. Very small energies can round to zero at the displayed precision.
Why is there no radius input?
Substituting I = βmR² and ω = v/R into ½Iω² cancels R². For fixed mass, speed and inertia factor, the energy therefore does not depend separately on radius. Radius is still needed to find angular speed or to determine β from a known moment of inertia.
Can I use this calculator if the body is slipping?
No. When slipping occurs, v and ω are independent. Total kinetic energy is still ½mv² + ½Iω², but the rotational part cannot be inferred from translational speed using ω = v/R.
How does the inertia factor affect the energy split?
At nonzero speed and positive mass, rotational energy divided by translational energy equals β. The rotational share of total energy is β/(1 + β), and the translational share is 1/(1 + β). At rest, both energies are zero, so their measured ratio is undefined.
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