Article
Galileo slowed free fall on a board. Without friction a = g sin α; with μ you get a = g(sin α − μ cos α).
Part 7 of Physics without mysteries. Next: energy in motion.
On a frictionless incline the along-board component of weight gives a = g sin α. At α = 30° and g = 9.81 m/s² you get a = 4.905 m/s² — exactly half of g, because sin 30° = 1/2. Galileo used a gentle board to time motion more carefully than free fall allows.
With length L and a start from rest, t = √(2L/a) and vₖ = √(2aL). That is the same uniformly accelerated motion, only a comes from the angle.
With friction the model becomes a = g(sin α − μ cos α). Force and Newton II on the slope close in F = m·a; as α → 90° you recover free fall.
| Goal | Formula | Tool |
|---|---|---|
| a without friction | a = g sin α | incline-no-friction |
| t and vₖ with L | t = √(2L/a), vₖ = √(2aL) | Same + L |
| a with friction | a = g(sin α − μ cos α) | incline-with-friction |
| Force / Newton II | F = m·a | force-f-ma |
A body slides without friction down an incline α = 30° of length L = 5 m. Take g = 9.81 m/s². Find a, slide time and vₖ (start from rest).
Steps
Answer: a ≈ 4.91 m/s², t ≈ 1.43 s, vₖ ≈ 7.00 m/s
