Velocity is a vector, so it can change in two ways: in size or in direction. Acceleration and Its Sign includes both. A car rounding a curve at a steady speed is accelerating, because its direction keeps turning.
Key idea
An object moving at steady speed $v$ around a circle of radius $r$ accelerates toward the center with $a_c = \dfrac{v^2}{r}$.
Why it works
Take the velocity at two nearby moments and subtract them as vectors: the change points toward the center of the circle. How big is it? In time $\Delta t$ the velocity turns through the same angle as the radius, $\Delta\theta = v\Delta t/r$, so the change has size about $v\,\Delta\theta = v^2\Delta t/r$. Divide by $\Delta t$: $a = v^2/r$. Faster means the velocity is both bigger and turning sooner, which is why $v$ is squared.
Worked examples
Worked example: A car on a curve
A car rounds a curve of radius $50\ \text{m}$ at $15\ \text{m/s}$. Find its acceleration.
- $a_c = \dfrac{15^2}{50} = \dfrac{225}{50} = 4.5\ \text{m/s}^2$, toward the center of the curve.
Answer: $4.5\ \text{m/s}^2$
Worked example: From the period
A carousel horse $5\ \text{m}$ from the center goes around once every $10\ \text{s}$. Find its acceleration.
- Speed: $v = \dfrac{2\pi r}{T} = \dfrac{2\pi(5)}{10} \approx 3.14\ \text{m/s}$.
- $a_c = \dfrac{3.14^2}{5} \approx 1.97\ \text{m/s}^2$.
Answer: about $1.97\ \text{m/s}^2$
Worked example: Doubling the speed
A runner rounds a curve at $4\ \text{m/s}$, then later at $8\ \text{m/s}$ on the same curve. How does the centripetal acceleration change?
- $a_c = \dfrac{v^2}{r}$: the speed is squared and $r$ is unchanged.
- Doubling $v$ multiplies $a_c$ by $2^2 = 4$.
Answer: It becomes 4 times as large.
Common mistake
Thinking an object at constant speed can't be accelerating. Acceleration measures any change in velocity, and turning changes the velocity's direction. Check: if the path curves, there is an acceleration with a part toward the inside of the curve.
Try it in the Science Lab: Circles and Gravity: The Apple and the Moon, which compares the Moon's centripetal acceleration with an apple's fall, the comparison Newton made.