Tools · Free play

Energy Explorer

Pick a scene, change its numbers, and watch the energy move: kinetic energy, gravitational and elastic potential energy, and the energy friction takes away. Switch to the work ledger to see \(\colKE{T_1} + \sum U_{1\to2} = \colKE{T_2}\) hold at every instant.

Choose a scene

Scene

Energy picture

\(\colKE{\text{Energy bars}}\) show where the energy is at each instant, with a stacked total that stays level. The \(\colKE{\text{work ledger}}\) starts from the kinetic energy at release and adds the work of each force, bar by bar, to land on the kinetic energy now.

Explore

Figure E.1 The chosen scene. Press Play or drag the time slider, and change the numbers in the panel (the motion is recomputed at once). Arrows: \(\colKE{\text{velocity}}\) in green, the \(\colVg{\text{weight}}\) in blue, the normal force in black, \(\colF{\text{friction}}\) in orange, the \(\colVe{\text{spring force}}\) in violet and an \(\colP{\text{applied force}}\) in amber. On a block they are drawn as in a free-body diagram: the weight's arrowhead ends at the center of gravity, a push (the normal force, a spring, an applied force) ends on the face it acts on, and friction starts on the face it acts on.

Things to try

  1. The spring always gives back. On the smooth incline, the crate bounces back to exactly where it started: \(\colVe{V_e}\) fills and empties, and the total bar never moves. Add friction and each bounce is lower.
  2. Pushing against friction. In the work ledger, \(\colP{U_P}\) stops growing once the push ends, while \(\colF{U_f}\) keeps growing until the crate stops and \(\colKE{T}\) is back to zero, so \(\colF{U_f} = -\colP{U_P}\).
  3. Just making the loop. Set the start height to \(2.5R\): the normal force at the top drops to almost zero. Go a little lower and the cart leaves the track and falls inside the loop.
  4. When does the string go slack? Give the bob a push at the start. Too little and it swings; too much and it goes over the top; in between, the string goes slack part way up and the bob falls inside the circle.
  5. Spring to height. With no friction, the launcher's block rises to \(h = \tfrac12 ks^2/mg\). Double the compression: the height goes up four times.
  6. Where the energy goes. On the coaster, watch the \(\colF{\text{lost}}\) bar grow fastest in the valleys, where the normal force, and so the friction, is largest.
  7. Leaving the dome. From a near standstill at the top, the block always leaves at \(\cos\theta = \tfrac23\) (about \(48.2^\circ\)), whatever the radius. Push it harder and it leaves sooner.

How the explorer works

Every scene is a particle on a track. The explorer integrates Newton's second law along the track in small time steps (fourth-order Runge–Kutta, \(\Delta t = 1\ \text{ms}\)), with the weight, the normal force \(N = m\left(\kappa v^2 + g\cos\phi\right)\) from the curvature \(\kappa\) of the track, kinetic friction \(\mu_k N\) against the sliding, linear springs and a constant applied force. Where a track can only push and \(N\) would have to pull, the particle leaves it and flies as a projectile until it meets the track again.

The energies are then computed from the motion, not assumed: \(\colKE{T = \tfrac12 mv^2}\), \(\colVg{V_g = Wy}\) from the datum, \(\colVe{V_e = \tfrac12 ks^2}\), and the works of friction and of \(P\) added up step by step. The total \(\colKE{T} + \colVg{V_g} + \colVe{V_e} - \colF{U_f} - \colP{U_P}\) stays equal to its starting value to within a small fraction of a percent, which is the principle of work and energy checked numerically.