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Physics C: Mechanics Unit 2: Newton's Laws, Friction and Drag

Newton's laws connect forces to motion. This unit builds the free-body-diagram method and then applies it to friction, circular motion and velocity-dependent drag, which needs a differential equation.

Covers: Newton's three laws and free-body diagrams, Friction: static and kinetic, Inclines and connected objects, Circular motion and banked curves, Linear drag and the time constant, Quadratic drag and terminal velocity, Time-varying forces

Study guide

Newton's first law says an object keeps a constant velocity unless a net force acts, so zero net force means zero acceleration, not zero velocity. The second law, ΣF = ma, is a vector equation: write it separately for each axis. The third law says forces come in pairs that act on different objects, are equal in size, opposite in direction and the same type. The core method is the free-body diagram: isolate one object, draw every force on it (weight, normal, tension, friction, drag, applied pushes), choose axes, often along and perpendicular to the acceleration, and write ΣF = ma on each axis.

Friction opposes sliding between surfaces. Static friction takes whatever value is needed to prevent slipping, up to fₛ ≤ μₛN. Once sliding starts, kinetic friction has the fixed value fₖ = μₖN, usually smaller than the static maximum. The normal force is not always mg: on an incline it is mg cos θ, in an accelerating elevator it is m(g + a), and a rope pulling upward at an angle reduces it. For connected objects, such as blocks linked by a string, treat the whole system to find the shared acceleration, then isolate one object to find an internal force like tension or a contact force.

An object moving in a circle at speed v has centripetal acceleration v²/r toward the center. Centripetal force is not a new kind of force; it is the net inward component of real forces like tension, friction, gravity or normal force. On a flat curve friction provides it, so v_max = √(μₛgr). On a banked curve with no friction the horizontal part of the normal force provides it, giving tan θ = v²/(rg). In a vertical circle, the tension or normal force changes around the loop because gravity sometimes points toward the center and sometimes away.

Drag forces depend on velocity, so the second law becomes a differential equation. For linear drag F = −bv on a falling object, m dv/dt = mg − bv. Separating variables gives v(t) = v_t(1 − e^(−t/τ)), where the terminal speed is v_t = mg/b and the time constant is τ = m/b. After one τ the object reaches 63% of v_t, and after about 4.6τ it reaches 99%. For an object coasting horizontally, m dv/dt = −bv gives v = v₀e^(−t/τ), and it travels a finite total distance mv₀/b. Quadratic drag, F = cv², fits faster, larger objects; its terminal speed is √(mg/c).

A force that changes with time must be integrated. If F(t) is known, a(t) = F(t)/m, and the velocity change is ∫a dt. Do not plug the final force into a constant-acceleration formula. The same thinking explains why terminal speed is approached from either side: if an object moves faster than v_t, drag exceeds weight and slows it down. With drag, a thrown ball rises more quickly than it falls, because drag adds to gravity on the way up and opposes it on the way down, and it returns with a lower speed than it was launched with.

Key ideas

• Net force, not velocity, determines acceleration.
• Third-law pairs act on different objects and never cancel each other.
• Static friction adjusts up to μₛN; kinetic friction is μₖN.
• Centripetal force is the net inward component of real forces.
• Linear drag gives exponential approach to terminal speed with time constant m/b.
• Terminal speed occurs when drag equals weight.
• Treat a system to find a shared acceleration, then isolate one object for internal forces.

Formulas and rules

Common mistakes

• Assuming the normal force always equals mg.
• Naming the normal force as the third-law partner of weight.
• Setting static friction equal to μₛN when the object is not on the verge of slipping.
• Adding a separate 'centripetal force' to the free-body diagram.
• Treating a drag or time-varying force as constant and using constant-acceleration equations.

Before the test, make sure you can

• Draw a correct free-body diagram and write ΣF = ma on each axis.
• Identify third-law pairs correctly.
• Solve incline and friction problems, including finding μₛ from a slipping angle.
• Solve connected-object problems for acceleration and tension.
• Analyze flat, banked and vertical circular motion.
• Derive and use v(t) for linear drag, including terminal speed and time constant.
• Calculate terminal speed for quadratic drag.
• Integrate a time-varying force to find velocity.

Words to know

Inertia

The tendency of an object to resist changes in its velocity; measured by mass.

Net force

The vector sum of all forces acting on an object.

Free-body diagram

A sketch of one isolated object showing every external force acting on it.

Normal force

The perpendicular contact force a surface exerts on an object.

Tension

The pulling force transmitted through a string, rope or cable.

Static friction

Friction that prevents surfaces from sliding; up to μₛN.

Kinetic friction

Friction between sliding surfaces; equal to μₖN.

Apparent weight

The normal force or scale reading, which differs from mg when the object accelerates vertically.

Centripetal acceleration

Acceleration v²/r toward the center of a circular path.

Banked curve

A tilted road where the normal force helps supply the centripetal force.

Linear drag

A resistive force proportional to speed, F = −bv.

Quadratic drag

A resistive force proportional to speed squared, F = cv².

Terminal velocity

The constant speed at which drag balances the driving force.

Time constant

τ = m/b for linear drag; the time to reach 63% of terminal speed.

Conical pendulum

A mass on a string moving in a horizontal circle, with the string tracing a cone.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.