Physics 1 (Calculus-Based) / Unit 103

What you'll learn

• Differentiate a circular position vector r(t) = R(cos ωt, sin ωt) twice to derive both the tangential velocity and the centripetal acceleration, rather than starting from a memorized formula
• Distinguish tangential acceleration (dv/dt, from changing speed) from centripetal acceleration (v²/R, from changing direction), and combine them as perpendicular vector components for non-uniform circular motion
• Differentiate and integrate angular position, velocity, and acceleration (θ, ω, α) using exactly the same techniques as linear kinematics
• Combine Newton's law of gravitation with the centripetal force requirement to find orbital speed and period

1. Deriving Centripetal Acceleration from a Position Vector

For an object moving on a circle of radius \(R\) at constant angular speed \(\omega\), its position can be written as a vector, \(\vec{r}(t) = R\langle\cos\omega t, \sin\omega t\rangle\). Differentiating once gives velocity; differentiating again gives acceleration — and the result isn't a separate fact to memorize, it falls directly out of differentiating this one expression twice.

2. Non-Uniform Circular Motion: Combining Tangential and Centripetal Acceleration

The derivation above assumed constant angular speed. The moment speed changes along the circular path, there's a second acceleration component — tangential acceleration, \(a_t = dv/dt\), pointing along the direction of motion (speeding up) or against it (slowing down) — in addition to the centripetal acceleration, \(a_c=v^2/R\), which is always present and always points toward the center. These two components are perpendicular to each other, so the total acceleration magnitude combines them the way the legs of a right triangle combine into a hypotenuse.

3. Angular Kinematics

Angular position, velocity, and acceleration relate to each other by differentiation and integration in exactly the same way as their linear counterparts from Unit 101: \(\omega(t) = d\theta/dt\), \(\alpha(t) = d\omega/dt\), and going the other direction, integrating \(\alpha(t)\) (with an initial condition) recovers \(\omega(t)\), and integrating that recovers \(\theta(t)\). Every technique from Unit 101 — including handling non-constant angular acceleration, which an algebra-based course can't do — carries over directly.

4. Gravitation and Circular Orbits

For an object in a circular orbit, gravity itself supplies the centripetal force — there's no separate force holding it in orbit. Setting Newton's law of gravitation equal to the centripetal force requirement, \(\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}\), and solving for \(v\) gives the orbital speed directly, with the orbiting object's own mass canceling out entirely.

Key equations

• r(t) = R(cos ωt, sin ωt) → a(t) = −ω²r(t) — Differentiating a circular position vector twice shows acceleration points toward the center with magnitude ω²R — this is where the centripetal acceleration formula actually comes from.
• a_tangential = dv/dt — The component of acceleration from changing speed along the circular path — zero only for uniform circular motion.
• a_centripetal = v²/R — The component of acceleration from changing direction, always present in circular motion, always pointing toward the center.
• |a_total| = √(a_t² + a_c²) — Tangential and centripetal acceleration are perpendicular, so the total acceleration magnitude combines them like the legs of a right triangle.
• ω(t)=dθ/dt, α(t)=dω/dt=d²θ/dt² — Angular kinematics — differentiate/integrate exactly as with linear position, velocity, and acceleration.
• v_orbit = √(GM/r) — Circular orbit speed, from setting gravitational force equal to the required centripetal force.

Open interactive practice for this unit