Newton's Second Law on a Slope
Resolving weight along and perpendicular to a slope turns an inclined-plane problem into one equation of motion plus one balance. Friction enters as a force set by the normal contact force and the surfaces in contact.
What a learner can do afterwards
- Resolves weight into components along and perpendicular to the surface
- Balances the perpendicular direction to get the normal contact force, then applies F = ma along the slope
- Uses the coefficient of friction to find the friction force and the angle at which sliding begins
1 · Read
Tilt the axes along and across the slope and weight is the only force needing a split. Weight m g becomes m g sin down the slope and m g cos into the surface. A 2 kg crate on a 30 degree slope with g 9.8 pulls 9.8 N down the slope and about 17.0 N into it.
Across the slope the forces balance, so the normal force equals the into slope part: N is m g cos. Along the slope apply F is m a to the net pull. Without friction a 2 kg crate on a 30 degree slope accelerates at g sin 30, which is 4.9 metres per second squared, and its normal force is about 17.0 N.
Friction is capped by mu times the normal force, and sliding starts when the down slope pull first beats that cap, giving tan theta is mu. With mu 0.4 the slide starts at about 22 degrees. A 2 kg crate on a 30 degree slope with mu 0.2 feels friction 3.4 N against a 9.8 N pull, so the net 6.4 N gives a of 3.2 metres per second squared.
Tilt axes along and across the slope every time: it keeps one equation of motion plus one balance. Steeper means a bigger down slope pull but a smaller normal force, since cos shrinks as the angle grows.
Split weight with sin down and cos in, balance across for N, then run F is m a along.
More to read
4.5 Normal, Tension, and Other Examples of Forces
Forces on a slope sort themselves once you tilt the axes. Point one axis down the surface and the other into it, and every force except weight already lies on an axis. Weight alone splits into mg sin theta along the slope and mg cos theta pressing in. The across direction then balances to give the normal force directly.
4.7 Further Applications of Newton’s Laws of Motion
Full slope problems run a fixed drill. Split the weight, balance across to find the normal, cap friction with mu times that normal, then apply F = ma along the surface. Static friction holds the block until the down slope pull outgrows the cap; past that, kinetic friction takes over and the block accelerates. Sliding starts where mg sin theta first beats mu mg cos theta, which simplifies to tan theta equals mu.
5.1 Friction
Friction deserves its own respect because it decides whether anything moves at all. It always opposes the slide that would happen without it, and it caps at mu times the normal force. On a slope the normal shrinks as the tilt steepens, so the friction cap shrinks too while the down slope pull grows. That double squeeze is why a gentle extra tilt can suddenly set a resting block gliding. Test grip questions by comparing the pull with the cap.
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2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.