What is centripetal acceleration?

Can an object accelerate if it's moving with constant speed? Yup! Many people find this counter-intuitive at first because they forget that changes in the direction of motion of an object—even if the object is maintaining a constant speed—still count as acceleration.

Acceleration is a change in velocity, either in its magnitude—i.e., speed—or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the speed might be constant. You experience this acceleration yourself when you turn a corner in your car—if you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion. What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. In this section we'll examine the direction and magnitude of that acceleration.

The figure below shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation—the center of the circular path. This direction is shown with the vector diagram in the figure. We call the acceleration of an object moving in uniform circular motion—resulting from a net external force—the centripetal acceleration $a_c$a, start subscript, c, end subscript; centripetal means “toward the center” or “center seeking”.

The direction of centripetal acceleration is toward the center of the circle, but what is its magnitude? Note that the triangle formed by the velocity vectors and the triangle formed by the radii $r$r and $\Delta s$delta, s are similar. Both the triangles $ABC$A, B, C and $PQR$P, Q, R are isosceles triangles with two equal sides. The two equal sides of the velocity vector triangle are the speeds $v_1=v_2=v$v, start subscript, 1, end subscript, equals, v, start subscript, 2, end subscript, equals, v. Using the properties of two similar triangles, we obtain $\dfrac{\Delta v}{v}=\dfrac{\Delta s}{r}$start fraction, delta, v, divided by, v, end fraction, equals, start fraction, delta, s, divided by, r, end fraction.

Acceleration is $\dfrac{\Delta v}{\Delta t}$start fraction, delta, v, divided by, delta, t, end fraction, so we first solve the above expression for $\Delta v$delta, v:

If we divide both sides by $\Delta t$delta, t we get the following:

Finally, noting that $\dfrac{\Delta v}{\Delta t}=a_c$start fraction, delta, v, divided by, delta, t, end fraction, equals, a, start subscript, c, end subscript and that $\dfrac{\Delta s}{\Delta t}=v$start fraction, delta, s, divided by, delta, t, end fraction, equals, v, the linear or tangential speed, we see that the magnitude of the centripetal acceleration is $a_c=\dfrac{v^2}{r}$a, start subscript, c, end subscript, equals, start fraction, v, squared, divided by, r, end fraction.

This is the acceleration of an object in a circle of radius $r$r at a speed $v$v. So, centripetal acceleration is greater at high speeds and in sharp curves—smaller radii—as you have noticed when driving a car. But it is a bit surprising that $a_c$a, start subscript, c, end subscript is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/hr than at 50 km/hr. A sharp corner has a small radius, so $a_c$a, start subscript, c, end subscript is greater for tighter turns, as you have probably noticed.

A centrifuge is a rotating device used to separate specimens of different densities. High centripetal acceleration significantly decreases the time it takes for separation to occur and makes separation possible with small samples. Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules—such as DNA and protein—from a solution.

Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity, $g$g; maximum centripetal acceleration of several hundred thousand $g$g is possible in a vacuum. Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth’s gravity.

What is the magnitude of the centripetal acceleration of a car following a curve, see figure below, of radius 500 m at a speed of 25 m/s—about 90 km/hr? Compare the acceleration with that due to gravity for this fairly gentle curve taken at highway speed.

Calculate the centripetal acceleration of a point 7.5 cm from the axis of an ultracentrifuge spinning at $7.5 \times 10^4$7, point, 5, times, 10, start superscript, 4, end superscript revolutions per minute.