KAHIBARO
Table of Contents
Direction of Acceleration in Circular Motion
When an object moves in a circle, its velocity is constantly changing, even if its speed stays the same. The reason is that velocity has both magnitude and direction. In circular motion, the direction of motion changes at every instant, so the object must have an acceleration.
This acceleration points toward the center of the circle. Because of this inward direction, it is called centripetal acceleration. The word "centripetal" means "center-seeking".
If an object is moving around a circle of radius $r$, then at any point its velocity is tangent to the circle, while its centripetal acceleration points directly inward.
Formula for Centripetal Acceleration
The magnitude of centripetal acceleration is
$$
a_c = \frac{v^2}{r}
$$
where $v$ is the speed of the object and $r$ is the radius of the circular path.
If the angular velocity $\omega$ is known, then using $v = \omega r$, we can also write
$$
a_c = \omega^2 r
$$
Both formulas describe the same inward acceleration.
For circular motion, the centripetal acceleration always points toward the center of the circle.
$$
a_c = \frac{v^2}{r} = \omega^2 r
$$
What the Formula Means
The formula $a_c = \frac{v^2}{r}$ shows two important ideas. First, if the speed increases, the centripetal acceleration increases very quickly because speed is squared. Second, if the radius becomes larger, the centripetal acceleration becomes smaller.
This means a sharp turn, which has small $r$, requires a larger inward acceleration than a wide turn.
Uniform Circular Motion
In uniform circular motion, the speed remains constant, but the direction changes continuously. So even though there is no change in speed, there is still acceleration. This is often surprising to beginners because acceleration is not only about speeding up or slowing down, it is also about changing direction.
In this case, the acceleration is purely centripetal.
Relation to Force
An acceleration requires a net force. So if an object moves in a circle, some force must act toward the center to produce the centripetal acceleration. The detailed discussion of forces belongs elsewhere, but the main connection is simple:
$$
F_{\text{center}} = m a_c = m \frac{v^2}{r}
$$
This inward force is not a new kind of force by itself. It is the role played by whatever real force points toward the center, such as tension, gravity, friction, or the normal force.
"Centripetal force" is the net inward force that causes circular motion. It is not a separate fundamental force.
$$
F_c = m \frac{v^2}{r}
$$
Everyday Examples
A car turning on a curved road needs centripetal acceleration toward the center of the curve. Friction between the tires and the road provides this inward effect.
A stone tied to a string and swung in a circle has centripetal acceleration toward the hand. The string tension provides the needed inward force.
The Moon moving around Earth also has centripetal acceleration toward Earth. In that case, gravity provides the inward force.
Comparing Speed, Radius, and Acceleration
The dependence of centripetal acceleration on speed and radius can be summarized clearly.
| Quantity change | Effect on $a_c$ |
|---|---|
| Double $v$ | $a_c$ becomes 4 times larger |
| Triple $v$ | $a_c$ becomes 9 times larger |
| Double $r$ | $a_c$ becomes half as large |
| Halve $r$ | $a_c$ becomes twice as large |
A Simple Numerical Example
Suppose a bicycle moves in a circular path of radius $5 \, \text{m}$ with speed $10 \, \text{m/s}$. Then
$$
a_c = \frac{v^2}{r} = \frac{(10)^2}{5} = \frac{100}{5} = 20 \, \text{m/s}^2
$$
So the bicycle has an inward acceleration of $20 \, \text{m/s}^2$.
If the mass of the bicycle and rider together is $80 \, \text{kg}$, then the required inward net force is
$$
F_c = m a_c = 80 \times 20 = 1600 \, \text{N}
$$
Key Idea
Centripetal acceleration is the acceleration required to keep an object moving along a circular path. Its direction is always inward, toward the center of the circle, and its magnitude is
$$
a_c = \frac{v^2}{r}
$$
Without this inward acceleration, the object would not continue in a circle. It would move off along a straight-line path tangent to the circle.