AP Physics 1 Unit 2 Notes: Circular Motion and the Radial Force Model

Circular motion

Motion in which an object’s path curves around a center point (often approximated as a circle).

Uniform circular motion (UCM)

Circular motion at constant speed; the object is still accelerating because its velocity direction continuously changes.

Velocity (in circular motion)

A vector tangent to the circle at every instant; its direction changes continuously even if speed is constant.

Radius (r)

Distance from the center of the circular path to the object.

Period (T)

Time for one full revolution (one complete trip around the circle); units: seconds.

Frequency (f)

Number of revolutions per second; units: hertz (Hz); related by f = 1/T.

Circumference

Distance traveled in one revolution of a circle: 2πr.

Tangential speed (v)

Speed along the circular path (tangent to the circle); for UCM, v = (2πr)/T.

Angular speed (ω)

Rate of rotation in rad/s; ω = 2π/T and it connects to linear speed via v = ωr.

Centripetal acceleration (a_c)

Inward acceleration required for circular motion; magnitude a_c = v²/r (points toward the center).

Centripetal acceleration using period

Form of centripetal acceleration when period is known: a_c = (4π²r)/T².

Radial (inward) direction

Direction from the object toward the center of the circular path; the centripetal acceleration points this way.

Tangential direction

Direction along the circle (tangent to the path); tangential acceleration points here when speed changes.

Perpendicular relationship in UCM

In uniform circular motion, centripetal acceleration is perpendicular to the velocity at every instant.

Centripetal force (concept)

Not a new type of force; it is the name for the net force directed toward the center that causes centripetal acceleration.

Radial force equation (Newton’s 2nd law)

Sum of real forces in the radial direction equals m(v²/r): ΣF_radial = m(v²/r).

Free-body diagram (FBD) for circular motion

Diagram showing only real forces (e.g., mg, N, T, friction); do not draw “centripetal force” or m(v²/r) as extra forces.

Conical pendulum

A mass on a string moving in a horizontal circle while the string makes an angle θ from vertical; tension provides both vertical support and radial inward force.

Conical pendulum component equations

Vertical: Tcosθ = mg; radial inward: Tsinθ = m(v²/r); combining gives tanθ = v²/(rg).

Apparent weight

The normal force you feel from a seat or scale; in vertical circular motion it changes with position and speed.

Bottom of a vertical circle (normal force)

At the bottom, inward is upward; radial equation: N − mg = m(v²/r), so N = mg + m(v²/r).

Top of a vertical circle (normal force)

At the top, inward is downward; radial equation: mg + N = m(v²/r), so N = m(v²/r) − mg.

Just-contact condition (top of hill/loop)

The condition for barely maintaining contact is N = 0; then mg = m(v²/r) and v_min = √(rg).

Maximum speed on a flat (unbanked) curve

When static friction provides the inward force: vmax = √(μs g r) (from m(v²/r) ≤ μ_s mg).

Non-uniform circular motion

Circular motion with changing speed; acceleration has two perpendicular components: radial ac = v²/r and tangential at = Δv/Δt; total magnitude a = √(ac² + at²).