Predict → set conditions → observe → record → change one variable. Loading an example clears records and creates a reproducible paused state.
01 · Same period, different speeds
Two disc points turn through the same angle in the same time, but arc length scales with radius.
v=r|\omega|,\qquad T=\frac{2\pi}{|\omega|}
Try itSet A at 2 m, B at 1 m and 60°/s; advance in one-second intervals. You can also drag markers to change radius.
Prediction. Both periods are 6 s; speeds are 2.094 m/s for A and 1.047 m/s for B.
Compare and explain. Why does the faster outer point not overtake the inner point?
Model limit. Markers are fixed on the same disc; radial slipping is omitted.
02 · Constant speed, nonzero acceleration
Velocity includes direction. Its tangent direction continually changes, requiring inward acceleration.
\vec a=-\omega^2\vec r,\qquad a=\frac{v^2}{r}
Try itShow velocity and acceleration; pause at the right, top and left of the circle.
Prediction. For r=2 m and 90°/s, speed is 3.142 m/s and acceleration magnitude is 4.935 m/s².
Compare and explain. What does perpendicular velocity and acceleration imply about speed changes?
Model limit. Do not compare lengths across different vector units; compare directions and like quantities.
03 · A finite change in velocity
Move two velocity vectors to a common tail to see their difference point inward.
\vec a_\mathrm{avg}=\frac{\vec v(t+\Delta t)-\vec v(t)}{\Delta t}
Try itUse Velocity change with Δt=0.50 s, then reduce the interval to 0.05 s in advanced settings.
Prediction. Average acceleration points toward the interval midpoint direction and approaches the initial instantaneous acceleration as Δt shrinks.
Compare and explain. Why is the finite-difference vector not exactly inward from the initial position?
Model limit. The difference is a velocity change, not a force; its unit is m/s.
04 · Which variable is fixed?
The effect of radius depends on whether angular speed or linear speed is held fixed.
a=r\omega^2=\frac{v^2}{r}
Try itRecord r=1 m at 60°/s. Record r=2 m at the same angular speed, then lower angular speed to 30°/s to keep linear speed fixed.
Prediction. Doubling radius doubles acceleration at fixed angular speed, but halves it at fixed linear speed.
Compare and explain. Explain the apparent contradiction between the two formulas using controlled variables.
Model limit. A parameter change resets time, separating states with different conditions.
05 · Project a circle onto one axis
A circular-motion component is harmonic. Position and acceleration have opposite signs; velocity differs by a quarter period.
x=r\cos\omega t,\quad v_x=-r\omega\sin\omega t,\quad a_x=-\omega^2x
Try itIn Projection view, select position, velocity and acceleration graphs. Compare t=0, 1 and 2 s.
Prediction. For r=2 m at 90°/s the period is 4 s, with x=2, 0, −2 m at those times.
Compare and explain. Why is speed zero while acceleration magnitude is largest at an endpoint?
Model limit. The projected point has variable speed; distinguish it from the point on the circle.
06 · Reverse the rotation
Reversing rotation reverses velocity, but at the same position centripetal acceleration is unchanged.
\vec v=\omega(-y,x),\qquad\vec a=-\omega^2(x,y)
Try itAt t=0 record +60°/s, then −60°/s. Compare velocity and the component graph phases.
Prediction. Speed, acceleration magnitude and period match; velocity components reverse signs.
Compare and explain. Why is the period positive even though angular velocity is signed?
Model limit. This compares separate initial conditions, not the force during an instantaneous reversal.
07 · Is centripetal force a new force?
Centripetal force names the inward net-force role. Maintaining the same motion requires force proportional to mass.
F_\mathrm{radial}=m\frac{v^2}{r}
Try itAt r=2 m and 90°/s compare masses of 1 kg and 2 kg, recording radial force.
Prediction. Motion is unchanged; required radial force doubles from 4.935 N to 9.870 N.
Compare and explain. With a limited friction force, which settings would cause slipping first?
Model limit. The model computes required force; it does not simulate a friction limit or slipping.
08 · Axis and zero rotation
A zero-radius point remains at the axis. Zero angular velocity means rest rather than circular motion.
r=0\Rightarrow v=a=0;\qquad\omega=0\Rightarrow T\to\infty
Try itSet radius A to zero and compare with B; then set angular velocity to zero.
Prediction. Speed and acceleration at the axis are zero. At zero angular velocity the period is shown as infinite.
Compare and explain. Can a motionless point reveal a measurable rotation direction or period?
Model limit. Use rω² at the axis rather than directly dividing v² by zero.