Rolling Without Slipping Calculator

Calculate linear velocity, rotational speed, and energy for rolling objects.

Please check your inputs and try again.

Rolling Without Slipping Couples Translation and Rotation

Pure rolling imposes the kinematic condition vCM=ωR, linking the center-of-mass speed to angular speed. The point of contact with a stationary surface has zero instantaneous velocity relative to the ground in ideal pure rolling. The top point moves at 2vCM. Total kinetic energy is the sum of translational and rotational parts: K=½Mv²+½Iω².

Static friction can be present even though the contact point does not slide. It may provide the torque needed for rolling acceleration, but for ideal rolling on a fixed surface static friction itself does no work at the instantaneous contact point.

v=ωR,   a=αR,   K=½Mv2+½Iω2
SymbolMeaningWhy it appears / units
vCenter-of-mass speedm/s.
ωAngular speedrad/s.
RRolling radiusm.
IMoment of inertia about centerkg·m²; depends on object shape.

Different shapes accelerate differently down the same incline because their moments of inertia distribute kinetic energy differently. A solid sphere has a smaller I/(MR²) than a hoop and therefore reaches the bottom faster under ideal rolling conditions.

The bottom contact point is the key no-slip check. For pure rolling on a stationary surface, the translational velocity of the center and the rotational velocity at the bottom cancel instantaneously. If v and ωR differ, the model describes slipping rather than rolling without slipping.

Worked Examples

Example 1: Solid sphere rolling at v=5m/s, m=2kg, r=0.1m
KE_t=25J, KE_r=½×2/5×2×0.01×2500
Result: KE_rot=10J (28.6%), total=35J
Sphere: 71.4% translational, 28.6% rotational
Example 2: Rolling down h=2m incline: solid vs hollow cylinder
v_solid=√(2gh/(1+0.5))=1.633√(gh)
Result: v_hollow=√(2gh/(1+1))=√(gh) — slower!
Solid cylinder beats hollow cylinder down ramp
Example 3: Wheel speed
R=0.30m, v=6m/s
Result: ω=20rad/s
Pure rolling fixes angular speed once radius and center speed are known.
Example 4: Top point speed
wheel center moves at 6m/s without slipping
Result: top point speed=12m/s
Rotational velocity adds to translational velocity at the top of the wheel.

Common Mistakes

⚠️
Assuming friction must be kinetic because the wheel moves

In pure rolling, the contact point is instantaneously at rest relative to the surface, so the relevant friction is static.

⚠️
Using v=ωR while the wheel is skidding

That condition is specifically the no-slip rolling constraint and fails during sliding.

⚠️
Ignoring rotational kinetic energy

A rolling object stores energy in both center-of-mass translation and rotation.

Frequently Asked Questions

Why do different shapes reach different speeds on inclines?
More rotational inertia (I/mr²) means more energy goes into rotation, less into translation. Hollow cylinder (k²=1) is slowest; solid sphere (k²=2/5) is fastest — confirmed by rolling race experiments.
Friction in rolling without slipping?
Static friction provides the torque to spin the object. No energy is lost to friction when rolling without slipping (static friction does no work). Energy is lost only if the object slips.
Why is the bottom point instantaneously at rest?
Its rotational velocity relative to the center is equal and opposite to the center’s translational velocity, so the two cancel at the contact point.
Why does the top point move at twice the center speed?
At the top, rotational velocity relative to the center points in the same direction as translational velocity, so v+v=2v.
Can static friction point downhill on a rolling object?
Yes, depending on applied torques and constraints. Static friction direction is determined by the tendency for relative slipping, not by a rule that it must oppose center-of-mass motion.
Which rolls faster down an incline, a hoop or solid sphere?
For equal size and mass under ideal no-slip rolling, a solid sphere accelerates faster because a smaller fraction of energy goes into rotation due to its lower I/(MR²).
What does v=ωR mean for rolling without slipping?
It states that the center-of-mass speed equals angular speed times radius when the contact point is instantaneously at rest relative to the surface. If the wheel skids, this constraint no longer holds. The total kinetic energy of a rolling rigid body includes both translation and rotation.

Formula Explorer connections

Interpretation: This formula is the rotational counterpart of linear mechanics, relating angle, angular motion, torque, inertia or rotational energy. Assumption: Define the rotation axis and sign convention. Rigid-body behavior, no slipping, steady rotation or negligible bearing losses may be assumed.

Shaft Torsion Calculator →Torque Calculator →Banked Curve Calculator →Physics Formula Explorer →