OPOpenPithMake curiosity visiblePhysics lab / Curved motion and energy

About this lab

Five experiments, two threads: how motion curves, and how energy moves.

The first three are about curved motion. A projectile splits into two straight-line motions; circular motion needs a net force toward the centre; for a satellite that force is gravity. The last two are about energy. Work is force accumulated over a displacement, power is how fast work is done, and at every step round the loop kinetic and potential energy trade places.

Every reading, dot and curve comes from the same physical model, not a pre-drawn animation. Change one parameter and all the data is recalculated together.

Five results

Five results

  • Projectile: flight time depends only on height, t = √(2H/g). From 0.8 m, whatever v0 is, ball A lands after 0.404 s, together with the dropped ball B.
  • Centripetal force: F = mω²r. Raising ω from 4 to 8 rad/s takes the force on a 0.2 kg slider at 0.15 m from 0.480 N to 1.920 N: four times, not twice.
  • Satellites: higher is slower. The space station (400 km) moves at about 7.67 km/s, one lap in 92 minutes; a geostationary satellite (35786 km) at about 3.07 km/s, one lap in 23.93 h. T²/r³ is the same for every orbit.
  • Work and power: raising a 0.5 kg block by 0.3 m on a smooth board takes different forces from 20° to 90°, but always 1.470 J of work. A ramp saves force, not work; pulling faster raises the power, not the work.
  • Roller coaster: with a 0.2 m loop, starting at 2R = 0.4 m never reaches the top; without friction it must start from at least 2.5R = 0.5 m. Starting at 0.45 m, the car leaves the track at about 146°.

Model and sources

Model and sources

  • Projectile: steady motion sideways, free fall downward; strobe every 0.05 s, grid read to 1 mm.
  • Centripetal force: the motor speeds up evenly to ω in 2 s, then holds for 3 s; the sensor reads F = mω²r, and the photogate recovers ω from the blocking time.
  • Satellites: circular orbits, GMm/r² = mv²/r; GM = 3.986 × 10¹⁴ m³/s², mean Earth radius 6371 km, sidereal day 86164 s. The BeiDou medium orbit is taken as about 21500 km.
  • Work and power: the winch pulls at constant speed, force = mg sinθ + μmg cosθ; W = Fs, P = Fv, efficiency η = mgh / W.
  • Roller coaster: the car is a point on the inside of the track, no rotation; in the loop N = m(v²/R + g cosφ), and when N reaches zero it leaves the track and flies as a projectile. Integrated in 0.2 ms steps.

g is the textbook value 9.8 m/s². The rough board’s friction coefficient of 0.3 and the coaster drag of 0.03mg are teaching assumptions for comparison, not measurements. Satellites are not to scale with Earth; orbit radii are. Arrow lengths are only proportional sketches.

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