In earlier chapters on kinematics, we learned that acceleration results from a change in velocity—either in magnitude, direction, or both. In uniform circular motion, an object moves at constant speed but continually changes direction. This change in direction means the object is always accelerating, even if its speed remains unchanged.

You may have experienced this type of acceleration while riding in a car that takes a sharp turn. Even if the car maintains a constant speed through the turn, you feel pushed sideways—this is a sensation caused by a change in the direction of motion, not speed. The tighter the curve or the faster the car, the more strongly this acceleration is felt.

Direction of Centripetal Acceleration

As shown in Figure 36.1, an object moving in a circle experiences a change in its velocity vector that always points toward the center of the circular path. This center-seeking acceleration is known as centripetal acceleration, denoted [latex]a_c[/latex] (from the Latin centrum petere, “to seek the center”).

Magnitude of Centripetal Acceleration

To derive the magnitude of [latex]a_c[/latex], we use geometric reasoning. Consider two positions of an object on a circular path and construct a triangle using the velocity vectors at those positions. Since both velocity vectors have the same magnitude [latex]v[/latex], this forms an isosceles triangle. A similar triangle can be drawn using the arc length [latex]\Delta s[/latex] and the radius [latex]r[/latex].

Using the similarity of the triangles, we find:

[latex]\frac{\Delta v}{v} = \frac{\Delta s}{r}[/latex]

Solving for [latex]\Delta v[/latex] and dividing by [latex]\Delta t[/latex]:

[latex]\Delta v = \frac{v}{r} \Delta s \quad \Rightarrow \quad \frac{\Delta v}{\Delta t} = \frac{v}{r} \cdot \frac{\Delta s}{\Delta t}[/latex]

Since [latex]\frac{\Delta s}{\Delta t} = v[/latex], we substitute to get:

[latex]a_c = \frac{\Delta v}{\Delta t} = \frac{v^2}{r}[/latex]

Thus, the magnitude of the centripetal acceleration is:

[latex]\boxed{a_c = \frac{v^2}{r}}[/latex]

This equation tells us that:

• Acceleration increases quadratically with speed: doubling the speed quadruples [latex]a_c[/latex],

• Acceleration increases inversely with the radius: tighter turns (smaller [latex]r[/latex]) result in larger [latex]a_c[/latex].

Centripetal Acceleration in Terms of Angular Velocity

Since the linear speed [latex]v[/latex] is related to the angular velocity [latex]\omega[/latex] by:

[latex]v = r\omega[/latex]

we substitute this into the expression for [latex]a_c[/latex]:

[latex]a_c = \frac{(r\omega)^2}{r} = r\omega^2[/latex]

So, we can also write centripetal acceleration as:

[latex]\boxed{a_c = r\omega^2}[/latex]

Use whichever form is more convenient—[latex]a_c = \frac{v^2}{r}[/latex] or [latex]a_c = r\omega^2[/latex]—depending on whether you are given linear or angular velocity.

Applications: The Centrifuge

A centrifuge (see Figure 36.2(b)) is a device that spins rapidly to generate very large centripetal accelerations. This acceleration allows materials to separate by density, mimicking the effect of gravity—but at much greater magnitudes (often hundreds of thousands of [latex]g[/latex]).

Centrifuges are widely used in:

• Biology – to separate blood cells, bacteria, or viruses from liquid media.

• Molecular biology – to isolate DNA, RNA, or proteins from solutions.

• Aerospace and medicine – in human centrifuges, used to test how astronauts or pilots respond to high [latex]g[/latex]-forces.

The ability to produce large centripetal accelerations allows centrifuges to achieve rapid and efficient separation with very small sample volumes, making them essential tools across many scientific and medical disciplines.

Centripetal Acceleration Requires a Net Force

As stated by Newton’s Second Law, a net external force is required to cause any acceleration. This principle applies to all forms of motion, including uniform circular motion, where the object’s speed is constant, but its direction is continuously changing, leading to centripetal acceleration.

Therefore, a net external force must be present to maintain circular motion. This force is always directed toward the center of the circle, causing the object to continuously deviate from a straight-line path.

In the next section—Centripetal Force—we will analyze the specific forces responsible for maintaining this motion in different physical contexts. Whether the force comes from tension, gravity, friction, or normal force, it must have a component pointing radially inward to sustain circular motion.

Without this net inward force, the object would move in a straight line, tangential to the circular path, in accordance with Newton’s First Law.