Foundations — Circular motion — centripetal acceleration derivation

This page assumes you have seen nothing. Every letter and squiggle the parent Circular motion — centripetal acceleration derivation note throws at you is unpacked here, in build-order, so that by the time you meet $a_c=v^2/r$ nothing is a stranger.

The building blocks, in order

1. Point, path, and circle

Look at the figure: the black dot is the centre, the blue line is one radius of length $r$, and the object (orange dot) rides on the rim. The radius is the same no matter where on the rim the object is — that constancy is what makes the maths clean later.

Why the topic needs it: the answer $a_c=v^2/r$ has $r$ in the denominator. A tighter circle (small $r$) means a sharper turn means more acceleration. You cannot read that formula without knowing what $r$ measures.

2. Vector — an arrow with size and direction

Why the topic needs it: the parent note's core sentence is "speed is constant but velocity changes." That sentence is only meaningful once you know a vector can change direction while keeping the same length.

3. Position vector $\vec{r}$ and velocity vector $\vec{v}$

In the figure the blue arrow $\vec{r}$ points outward to the object; the orange arrow $\vec{v}$ points along the rim — it is tangent to the circle.

Why the topic needs it: both derivation methods start by writing $\vec{r}$ and reading $\vec{v}$ off it. The fact that $\vec{v}\perp \vec{r}$ is the single hinge the geometry method turns on.

4. Speed $v$ vs. velocity $\vec{v}$

5. The change $\Delta$ (Greek "delta")

The figure shows the trick that makes the whole derivation work. Slide the old and new velocity arrows so their tails meet. The red arrow $\Delta \vec{v}$ that closes the gap — tip of old to tip of new — is the change in velocity. Notice it does not point along the motion; it leans inward. That inward lean, in the limit, becomes the centripetal acceleration.

Why the topic needs it: acceleration is defined as $\Delta \vec{v}/\Delta t$. Without $\Delta$ you cannot even write down what acceleration means.

6. Angle $\theta$ and the radian

7. Angular velocity $\omega$ and period $T$

Why the topic needs it: Worked Example 3 in the parent turns $\omega$ into $2\pi /T$ to get $a_c=4{\pi }^{2}r/T^2$. That step is invisible unless you know these two definitions. See Angular velocity and period.

8. The derivative $\frac{d}{dt}$ and the limit

Why this tool and not another? We need the exact acceleration at one instant, not an average over a visible slice of the path. Only the limit/derivative answers "rate of change right now."

See Vectors — derivative of a unit vector for the calculus route (Method 2), where differentiating $\vec{r}=r(\cos \omega t,\sin \omega t)$ twice pops out $-{\omega }^2\vec{r}$.

9. Acceleration $a_c$ and Newton's laws

The subscript $c$ in $a_c$ just labels it centripetal ("centre-seeking"): the acceleration that points inward. It exists because Newton's First Law (Inertia) says an object left alone travels straight — so bending it into a circle requires a real inward force, tied to $a_c$ through Newton's Second Law as $F_c=m{a}_c$.

How the foundations feed the topic

Read top to bottom: geometry (circle, vectors) builds the two arrows; $\Delta$ builds the inward change; radians and $\omega$ give the timing; the limit sharpens the approximation into an exact rate; Newton's laws explain why the inward acceleration must exist at all.

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