Physics I: Mechanics › Forces › full formula sheet

N = mg cos θSay it: “the normal force on an incline equals m g cosine theta”

Normal force on an incline

Why a slope pushes back with less than your full weight — and where the rest goes.

Notation on this page: N is the normal force in N, m is mass in kg, g = 9.8 m/s², θ is the incline angle above the horizontal.

Before this lesson: Weight, Newton's second law

Where it comes from

On flat ground the deal is simple: weight pulls down with mg, the floor pushes up with N = mg, done. Tilt the floor and the deal changes. Weight still points straight down — gravity doesn't tilt — but the surface can only push perpendicular to itself. Part of the weight now presses into the slope, and part drags the block along it.

Before reading on: a 10 kg block sits on a 30° ramp. Is the normal force 98 N (full weight), more than 98 N, or less? Guess before you compute.

Split the weight arrow into two components — one perpendicular to the slope, one parallel:

mg = (mg cos θ)⊥ + (mg sin θ)∥the perpendicular part squeezes the surface; the parallel part pulls downhillSay it: “weight splits into m g cosine theta perpendicular and m g sine theta parallel”

The surface answers only the perpendicular squeeze — so N = mg cos θ. For the 30° ramp: cos 30° ≈ 0.866, so N ≈ 84.9 N — less than the 98 N weight. The “missing” 49 N (= mg sin 30°) pulls the block downhill, which is exactly why ramps make things slide.

Derivation

This is Newton's second law wearing tilted axes. The trick: aim one axis perpendicular to the slope, where nothing moves, so the second law collapses to a balance.

ΣF⊥
=
ma⊥ = 0
Step 1 — no motion into the slope. The block doesn't sink through the surface, so a⊥ = 0. Perpendicular forces must balance exactly.
N − mg cos θ
=
0
Step 2 — list the perpendicular forces. N pushes out of the surface (+); the perpendicular component of weight, mg cos θ, presses in (−). (Geometry: the angle between the weight arrow and the normal direction equals the incline angle θ.)
N
=
mg cos θ
Step 3 — solve. The normal force equals the perpendicular component of weight. ∎

Reality-check the limits: θ = 0 (flat) → N = mg ✓; θ = 90° (vertical wall) → N = 0, and the block falls freely ✓. A formula that nails both extremes is one you can trust in between.

Why cosine? In the force triangle, the normal direction sits adjacent to the angle θ — and adjacent/hypotenuse is cosine. If you ever mix up sin and cos, the θ = 0 limit rescues you: flat ground must give N = mg, and only cosine does that.

How to use it

The procedure, every time:

  1. Draw weight straight down — always vertical, never perpendicular to the slope. This is the #1 drawing error.
  2. Tilt your axes (or resolve the weight): perpendicular component mg cos θ, parallel component mg sin θ.
  3. Read off N = mg cos θ from the perpendicular balance.
  4. Feed N into friction if needed: fk = μkmg cos θ. And the downhill motion comes from the parallel piece: a = g sin θ (frictionless).
Common mistake: writing N = mg on a slope. The steeper the ramp, the worse this gets — at 60° the true normal is only half the weight.

Worked examples

Four problems, easiest first. In each one, read every step — the why of each move is the lesson.

Example 1 — the basic move: 10 kg on a 30° incline

  1. Weight. mg = 10 × 9.8 = 98 N, straight down.
  2. Normal. N = mg cos 30° = 98 × 0.8660 ≈ 84.9 N.
  3. Downhill pull. mg sin 30° = 98 × 0.5 = 49 N — the part the normal force doesn't touch.
Common mistake: N = 98 N (“weight is weight”). On a slope only the perpendicular component counts — 13 N of your answer just slid downhill.
Your turn — 5 kg on a 30° incline. Normal force?

Answer: about 42.4 N. N = 49 × 0.8660 ≈ 42.43 ≈ 42.4 N.

Before reading on: same 10 kg block, but the ramp steepens to 60°. Does the normal force grow or shrink — and by roughly how much?

Example 2 — steeper: 10 kg on a 60° incline

  1. Normal. N = 98 × cos 60° = 98 × 0.5 = 49 N — exactly half the weight.
  2. Downhill pull. 98 × sin 60° = 98 × 0.8660 ≈ 84.9 N.
  3. The trade: steeper slope → smaller normal, bigger downhill pull. The components swap roles as θ grows.
Common mistake: using sin for the normal (“N = mg sin θ”). The θ = 0 test kills it: flat ground would give N = 0, absurd.
Your turn — 2 kg on a 45° incline. Normal force?

Answer: about 13.9 N. N = 19.6 × 0.7071 ≈ 13.86 ≈ 13.9 N.

Example 3 — frictionless slide: acceleration down a 30° ramp

  1. Parallel forces. Only mg sin θ pulls downhill (no friction); nothing opposes it.
  2. Second law along the slope. mg sin θ = ma → a = g sin θ.
  3. Compute. a = 9.8 × 0.5 = 4.9 m/s² down the slope — half of free fall, independent of mass.
Common mistake: answering 9.8 m/s² (“it's falling”). Only the parallel component accelerates the block — the perpendicular part is cancelled by the normal force.
Your turn — frictionless slide down a 45° ramp. Acceleration?

Answer: about 6.9 m/s². a = 9.8 × 0.7071 ≈ 6.93 ≈ 6.9 m/s² down the slope.

Example 4 — with friction: 4 kg, 30°, μk = 0.2

  1. Normal. N = mg cos 30° = 39.2 × 0.8660 ≈ 34.0 N.
  2. Friction. fk = μkN = 0.2 × 33.95 ≈ 6.8 N, up the slope (opposing the slide).
  3. Net downhill. mg sin 30° − fk = 19.6 − 6.79 = 12.81 N.
  4. Accelerate. a = 12.81/4 ≈ 3.2 m/s² down the slope (vs 4.9 frictionless).
Common mistake: fk = μkmg = 7.8 N — using the full weight instead of the reduced normal. On slopes, friction always shrinks with cos θ.
Your turn — 2 kg, 30°, μk = 0.1. Acceleration down the slope?

Answer: about 4.1 m/s². N ≈ 17.0 N; fk ≈ 1.70 N; net = 9.8 − 1.70 = 8.10 N; a = 8.10/2 = 4.05 ≈ 4.1 m/s².

Memorization tips

  • Say it aloud: “normal is m g cosine theta; downhill is m g sine theta.” The pair rhymes — learn them together.
  • The θ = 0 test: flat ground must give N = mg. Only cosine survives — your instant sin/cos tiebreaker, forever.
  • The θ = 90° test: vertical wall → N = 0, free fall. Both limits bracket the formula.
  • Draw weight vertical: gravity never tilts. Tilt the axes, not the weight arrow.
  • Friction feeds on N: on a slope, f = μmg cos θ. Steeper ramp → less normal → less grip — two reasons slopes are slippery.
  • a = g sin θ: the frictionless-slide shortcut. Mass cancels again — Galileo's ghost haunts every ramp.

Final challenge

Five mixed questions — components, limits, friction on slopes, and the traps, all in one. Score 5/5 and the incline is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

What is the normal force on an incline?

N = mg cos θ: on a slope of angle θ, the surface pushes back with only the perpendicular component of the weight. It is always less than mg (for 0 < θ < 90°).

Why is the normal force less than mg on a slope?

Weight points straight down, but the surface can only push perpendicular to itself. Only the perpendicular component (mg cos θ) presses into the surface; the rest (mg sin θ) pulls the object downhill.

Why cosine and not sine?

The normal direction is adjacent to the angle θ in the force triangle, and adjacent means cosine. Check the limits: θ = 0 gives N = mg (flat ground ✓), θ = 90° gives N = 0 (vertical wall ✓).

How fast does a block slide down a frictionless incline?

a = g sin θ down the slope — the parallel component of weight is the only unbalanced force. At 30° that is 4.9 m/s², half of free fall.

How does the incline change friction?

Friction scales with the normal force, so on a slope f = μN = μmg cos θ — less grip than on flat ground, which is why things slide more easily on steep ramps.

More from the codex

Live

Physics I formula sheet

All the mechanics formulas — kinematics, forces, energy — printable and quiz-ready.

Open sheet → Live

Formula Sheet Builder

Mix and match any sections into your own printable sheet.

Open tool → Live

Prompt Simulator

Practice prompt engineering with deterministic scoring.

Open tool →