Calculator · Engineering Mechanics
Rolling Without Slipping Calculator
Pure Rolling of an axisymmetric body on a flat surface. The No Slip constraint ties the translation of G to the spin, and on an incline it also fixes the static friction the contact must supply. This is not a general contact solver: sliding is out of scope.
Mode
Pure rolling kinematics
Positive in the direction of travel, taken as +x.
Results
| Relation used | ω = −v_G / R and α = −a_G,t / R |
|---|---|
| Rolling radius R | 0.5 m |
| Speed of G, v_G | 4 m/s |
| Angular velocity ω | -8 rad/s — clockwise |
| Tangential acceleration a_G,t | 0 m/s² |
| Angular acceleration α | 0 rad/s² |
| Velocity of the contact point C | (0, 0) m/s — from v_C = v_G + ω k × r_C/G |
| Speed of C | 0 m/s |
| Velocity of the top point T | (8, 0) m/s |
| Speed of T | 8 m/s — twice |v_G| for a wheel on a flat surface |
The contact and top velocities are evaluated with the same general rigid-body relation the Rigid-Body Kinematics engine uses, not from a separate rolling shortcut.
Technical reference
Signed flat-surface kinematics: v_G = −ω·R (⇔ |v_G| = R·|ω|), a_G,t = −α·R
Incline magnitudes: |α| = a_G / R, f_required = I_G·a_G / R²
- R
- rolling radius, contact point at (0, −R) from G[m]
- v_G, a_G,t
- signed flat-surface velocity and tangential acceleration of the centre of mass, +x positive[m/s, m/s²]
- a_G
- on the incline, the positive acceleration magnitude of G down the slope[m/s²]
- ω, α
- angular velocity and acceleration, CCW positive[rad/s, rad/s²]
- f_required
- magnitude of the static friction the no-slip condition demands, on the incline[N]
v_G = −ω·R and a_G,t = −α·R are the KINEMATIC CONSTRAINT: they describe pure rolling once it is assumed, and hold regardless of what supplies it. They are signed relations in the flat-surface frame, with travel along +x and counter-clockwise ω, α positive.
The incline section reports magnitudes instead: a_G is the positive acceleration down the slope, |α| = a_G/R, and f_required is a magnitude whose direction along the slope is stated separately in the results.
On the incline that assumption also fixes a required friction f = I_G·a_G/R² from ΣM_G = I_G·α with a_G = αR. That is a SUFFICIENT-FRICTION check, evaluated only when a coefficient of static friction μs is supplied — it compares f against μs·N and reports whether pure rolling is feasible. With no μs entered, the friction is reported but never checked.
Assumptions
- · The surface lies below the body, so the contact point is at r_C/G = (0, −R). With ω counter-clockwise positive, Pure Rolling means v_G = −ωR and a_G,t = −αR.
- · The No Slip constraint is assumed, not tested, unless a μs is supplied. When it is, the required friction is compared with μs·N and the verdict is stated.
- · The incline model is a single axisymmetric body on a rigid surface, with I_G about its symmetry axis through G.
- · Rolling resistance, deformation and a moving surface are not modelled.
- · If the required friction exceeds μs·N the body would slip, and a sliding model with kinetic friction is outside this tool — nothing switches over automatically.
Continue
The exact rolling state, with the contact point and the top point already placed as body points, so the zero contact velocity and the 2·v_G top speed can be read off the general relations.