2.6 Applications of Newton's Second Law

Setting up the problem

Applying Newton's Second Law starts with a free-body diagram identifying all forces on an object. Choose axes (often along/perpendicular to motion or an incline), sum forces in each direction, and set them equal to ma. For systems with multiple objects, apply ΣF = ma to each object separately, using Newton's Third Law pairs and shared acceleration/tension as links.

Inclines and friction

On an inclined plane, decompose gravity into components: mg sinθ (parallel to surface) and mg cosθ (perpendicular). Normal force balances the perpendicular component, and friction (f = μN) opposes relative motion or impending motion along the surface. Static friction adjusts up to μₛN to prevent sliding; kinetic friction is μₖN once sliding occurs.

Connected objects & pulleys

For objects connected by strings over pulleys (like Atwood machines), assume massless, inextensible strings and frictionless, massless pulleys unless stated otherwise. Tension is the same throughout a single string, and both masses share the same magnitude of acceleration. Write ΣF = ma for each mass, then solve the system simultaneously.

Key terms

Free-body diagram

A diagram showing all forces acting on a single object, used to apply Newton's Second Law.

Incline force components

Gravity splits into mg sinθ (parallel to surface, causes sliding) and mg cosθ (perpendicular, balanced by normal force).

Kinetic vs. static friction

Static friction (up to μₛN) prevents motion from starting; kinetic friction (μₖN) acts once objects are sliding, typically μₖ < μₛ.

Atwood machine

Two masses connected by a string over a pulley; shared tension and acceleration magnitude let you solve ΣF = ma for each mass simultaneously.

Apparent weight in accelerating elevator

N = m(g ± a); normal force increases when accelerating upward and decreases when accelerating downward.

Try an AP-style question

A block slides down a frictionless 30° incline. What is its acceleration? (g ≈ 10 m/s²)

1. 5 m/s²
2. 0 m/s²
3. 8.7 m/s²
4. 10 m/s²