Tools · Free play

Motion Explorer

Pick a motion, or type your own \(r(t)\), \(\theta(t)\) and \(z(t)\). Play it, scrub it, and watch the velocity, the acceleration and every one of their terms, in a top view or in 3D.

Choose a motion

Motion

View and mass

Explore

Figure E.1 The chosen motion. Press Play or drag the time slider; in the top view you can also drag \(P\) along its path. Green is the velocity \(\vvec\) and red the acceleration \(\avec\); orange and violet are their components along \(\er\) and \(\et\) (and blue along \(\ez\) in 3D). The live numbers list every term of the acceleration and the resultant force \(\sum\Fvec = m\avec\) that the motion needs.

Things to try

  1. Collar sliding out on a spinning rod. Both rates are constant, yet \(\avec \ne \mathbf{0}\). Pause and check \(a_r = -r\dot\theta^2\) and \(a_\theta = 2\dot r\dot\theta\) against the live numbers.
  2. Free collar on a driven rod. With nothing pushing along the rod, \(\sum F_r = 0\), so \(a_r = 0\) at every instant: \(\ddot r\) exactly cancels \(-r\dot\theta^2\). The rod still pushes sideways.
  3. Straight line at constant speed. \(\vvec\) never changes, yet \(v_r\) and \(v_\theta\) do, and so do the individual acceleration terms. They cancel exactly, so \(\avec = \mathbf{0}\).
  4. Puck pulled in on a cord. The only horizontal force points at \(O\), so \(a_\theta = 0\) and \(r^2\dot\theta\) stays at \(1.28\ \text{m}^2/\text{s}\) while the puck spins faster and faster.
  5. Pin on a cardioid cam. Turn on the angle \(\psi\) and watch it change around the cam; near the cusp the pin slows right down.
  6. Robot arm in 3D. Switch to Top view and back: \(z\) adds a component \(v_z\) but no acceleration, because \(\dot z\) is constant.
  7. Your own motion. Choose Custom and try a circle \(r = 1\), \(\theta = t\); then make it speed up with \(\theta = 0.5t^2\); then let it spiral out with \(r = 1 + 0.2t\).

How the explorer works

For every motion the explorer evaluates \(r(t)\), \(\theta(t)\) and \(z(t)\), finds the rates \(\dot r\), \(\ddot r\), \(\dot\theta\), \(\ddot\theta\), \(\dot z\), \(\ddot z\) by numerical differentiation (central differences), and then applies the formulas of Lessons 4 and 5:

\[ \vvec = \dot r\,\er + r\dot\theta\,\et + \dot z\,\ez, \qquad \avec = \left(\ddot r - r\dot\theta^2\right)\er + \left(r\ddot\theta + 2\dot r\dot\theta\right)\et + \ddot z\,\ez \]

The resultant force is \(\sum\Fvec = m\avec\) (Lesson 7). Values smaller than about \(10^{-6}\) are shown as 0, because numerical derivatives carry tiny rounding errors. Arrow lengths are scaled to fit; the note under the controls gives the scale.