Curvilinear motion follows a curved path. This interactive example moves a point mass around an ellipse at constant speed, showing how the tangent direction, local curvature radius and inward acceleration change along the path. Set equal semi-axes to obtain uniform circular motion.
Curvilinear Motion Interactive Calculator
Explore a point moving at constant speed along an elliptical path. Adjust the semi-axes, speed and mass; see local curvature, acceleration and required net force.
ρ = (a²sin²θ + b²cos²θ)^(3/2)/(ab); an = v²/ρ; Fnet = m an; T = path length/v
Circuit position follows numerical arc length. Normal acceleration varies with local curvature; tangential acceleration is zero.
- Planar elliptical point trajectory at constant speed.
- No gravity or external load included in net in-plane force.
- Independent display scales for velocity and acceleration arrows.
- A zero-speed input gives a stationary point and no finite circuit time.
Constant speed; point mass; no individual bearing or contact reaction model.
A moving frame follows the path
Velocity lies along the path tangent. Normal acceleration points toward the local center of curvature. At constant speed there is no tangential acceleration, even though the velocity vector changes direction.
The animation advances by arc length, so equal time intervals cover equal distances. Uniformly advancing an ellipse’s parameter angle would produce varying speed; this calculator does not make that substitution. The orange arrow follows the inward normal, which need not point toward the ellipse’s geometric center.
Explore a curved point path
Adjust the horizontal and vertical semi-axes to change the path. The display fits the whole ellipse into the window; the stated dimensions remain in metres. Speed changes acceleration quadratically, while mass scales the required net force without changing the geometry.
This is a point-path demonstration, not a model of a particular conveyor, cam follower or vehicle. Those systems need additional dimensions, loads and contact constraints before individual bearing or guide forces can be determined.
Ellipse curvature and constant-speed acceleration
For x=a cos θ and y=b sin θ, define q=√(a² sin²θ+b² cos²θ). The local curvature radius is ρ=q³/(ab). Normal acceleration is aₙ=v²/ρ and the required net in-plane force is F=m aₙ. Here tangential acceleration is zero.
The path length is the integral of q over one complete turn. The calculator evaluates it numerically, then uses cumulative arc length to locate the point. Circuit time is path length divided by speed. At zero speed the point remains stationary and no finite circuit time is reported.
For a general changing-speed path, total acceleration combines the tangential term dv/dt and normal term v²/ρ as perpendicular components. That broader case is explained here but is not simulated by this constant-speed example.
Two-metre by one-metre semi-axes
With a=2 m and b=1 m, the curvature radius is 0.5 m at the rightmost point and 4 m at the top. At 1 m/s, the corresponding normal accelerations are 2 and 0.25 m/s². For a 5 kg point mass, the required net in-plane forces are 10 and 1.25 N.
The path length is approximately 9.68845 m, giving a 9.68845-second circuit at 1 m/s. With equal 1 m semi-axes, the path becomes a circle: curvature radius stays 1 m, acceleration is 1 m/s² and circuit time is 2π seconds.
Net force is not a bearing specification
The calculated force is the resultant needed for the selected point path. It does not include gravity, friction, external loads, load sharing or moments. A bearing reaction may differ substantially from this resultant.
No universal radius tolerance, acceleration limit or bearing lifetime follows from these inputs. Material, contact geometry, lubrication, duty cycle and the complete force system must be assessed separately.
Curved-path questions
Why does force change at constant speed?
The ellipse’s local curvature changes. Smaller curvature radius requires larger normal acceleration at the same speed.
Why does the orange arrow not always point to the center of the ellipse?
Normal acceleration points toward the local curvature center, which generally differs from the ellipse center.
What does mass change?
It scales net force. The specified trajectory, speed and acceleration remain the same.
What happens at zero speed?
The point stops, velocity and acceleration arrows vanish, and circuit time reads Stationary.
Are arrows drawn to one common scale?
No. Velocity and acceleration have different units and use independent display scales.
References
OpenStax University Physics, Uniform and Nonuniform Circular Motion, for tangent and normal acceleration components.
OpenStax University Physics, Centripetal Force, for the relation between mass, normal acceleration and net force. The parametric equations above define the ellipse’s local curvature and arc length.
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