Forces in 2D; friction; circular motion - Physics C: Mechanics AP Study Notes

Forces in Two Dimensions require vector decomposition into perpendicular components (typically x and y). Any force F at angle θ becomes Fx = F cos θ and Fy = F sin θ. When multiple forces act, use ΣF = ma separately for each dimension: ΣFx = max and ΣFy = may.

Friction opposes relative motion between surfaces. Static friction (fs) prevents motion up to a maximum: fs ≤ μsN, where μs is the coefficient of static friction and N is the normal force. Once motion begins, kinetic friction (fk = μkN) acts opposite to velocity, with μk < μs typically. The normal force equals the perpendicular component of contact forces, not always equal to weight.

Circular Motion requires centripetal acceleration ac = v²/r = ω²r directed toward the center, where v is tangential speed, r is radius, and ω is angular velocity. The centripetal force Fc = mv²/r is not a new force type but the net force causing circular motion. This could be tension, gravity, friction, or combinations thereof.

Key equations:

• Newton's Second Law: ΣF = ma
• Friction: f ≤ μN (static), f = μN (kinetic)
• Centripetal: ac = v²/r, Fc = mv²/r
• Kinematics: v = ωr, ω = 2π/T (T = period)

Critical distinction: Centripetal force is the result of other forces, not a separate force to add to free-body diagrams.

2D Forces in Action: Consider an airplane ascending at 15° while experiencing lift (perpendicular to wings), thrust (forward), drag (backward), and weight (downward). Pilots must resolve these vectors to maintain stable climb angles. Engineers design wing angles knowing that lift's vertical component must exceed weight while its horizontal component affects turning radius.

Friction's Dual Nature: When you push a heavy box, static friction initially matches your push exactly—the box doesn't move until you exceed μsN. Think of static friction as a "responsive" force that adjusts from zero up to its maximum. Once sliding begins, kinetic friction becomes constant at μkN. This explains why starting a stalled car requires more torque than keeping it moving. Ice skating demonstrates ultra-low μk (~0.02), while rubber on dry concrete has μk ≈ 0.7.

Circular Motion Examples: A car turning at speed relies on friction as centripetal force. The maximum safe speed on a flat curve: v = √(μgr). Banking the road adds a component of normal force toward the center, allowing higher speeds. Satellites orbit Earth because gravity provides exactly the centripetal force needed: mg = mv²/r, giving orbital velocity independent of satellite mass. A roller coaster's loop requires careful engineering—at the top, normal force plus weight must equal mv²/r, so N = mv²/r - mg. If v is too low, N becomes negative (impossible), and riders fall.

Memory aid: "Friction Fights, Circular Centers"—friction opposes motion's direction; centripetal forces point to rotation's center.