Circular Motion and Gravitation (Newton’s Laws on Curved Paths)

Uniform circular motion

Motion along a circular path at constant speed; velocity changes because its direction continuously changes.

Velocity (in circular motion context)

A vector that is tangent to the circular path at each instant; not constant in circular motion because direction changes.

Centripetal acceleration

Inward (center-seeking) acceleration required to keep an object moving in a circle; points toward the center.

Radial direction

The direction toward or away from the circle’s center; used to analyze centripetal effects (often choose inward as positive).

Tangential direction

The direction along the instantaneous motion (tangent to the circle); associated with changes in speed.

Tangential speed (v)

The magnitude of the velocity for circular motion; related to angular speed by v = ωr.

Angular speed (ω)

Rate of change of angular position (in rad/s); related to period and frequency by ω = 2π/T = 2πf.

Period (T)

Time for one full revolution around the circle.

Frequency (f)

Number of revolutions per second; related to period by f = 1/T and to angular speed by ω = 2πf.

Centripetal acceleration magnitude (v-form)

a_c = v^2/r, where v is instantaneous tangential speed and r is the radius.

Centripetal acceleration magnitude (ω-form)

a_c = ω^2r, obtained using v = ωr.

Net radial force equation

Applying Newton’s 2nd law inward: ΣF_r = m(v^2/r) (or mω^2r), using only radial components of real forces.

“Centripetal force” (interpretation)

Not an extra force to draw; it is the name for the net inward (radial) force required for circular motion.

Static friction as centripetal force (flat curve)

On an unbanked turn, static friction can supply the inward net force needed to curve the motion.

Minimum coefficient of static friction for a flat turn

For speed v on radius r without slipping: μ_s ≥ v^2/(rg).

Conical pendulum

A mass on a string moving in a horizontal circle while the string makes an angle θ with the vertical; tension provides both support and centripetal effect via components.

Conical pendulum radius relation

For string length L and angle θ from vertical: r = L sinθ.

Non-uniform circular motion

Circular motion where speed changes with time; acceleration has both radial (direction change) and tangential (speed change) components.

Tangential acceleration (a_t)

Acceleration component that changes speed: a_t = dv/dt; points with motion when speeding up and opposite when slowing down.

Total acceleration in non-uniform circular motion

Because radial and tangential components are perpendicular: a = √(ar^2 + at^2).

Angular acceleration (α)

Rate of change of angular speed: α = dω/dt; connects to tangential acceleration by a_t = rα.

Vertical circular motion (radial force balance)

Circular motion in a vertical plane where gravity’s radial contribution changes with position, affecting tension/normal force around the loop.

Minimum speed at top of a vertical circle (string just taut)

At the top with T = 0, gravity alone supplies centripetal requirement, giving v_min = √(gr).

Newton’s law of universal gravitation

Attractive force between masses m1 and m2 separated by distance r: F_g = Gm1m2/r^2.

Circular orbit speed and period (gravity provides centripetal force)

For a circular orbit of radius r around mass M: v = √(GM/r) and T = 2π√(r^3/(GM)); r must be measured from the planet’s center (r = R + h).