PHYS101: Introduction to Mechanics

We know from kinematics that acceleration is a change in velocity, either in its magnitude 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 magnitude of the velocity 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 examine the direction and magnitude of that acceleration.

Figure 6.8 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 pointing 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($a_{c}$); centripetal means "toward the center" or "center seeking".

Figure 6.8 The directions of the velocity of an object at two different points are shown, and the change in velocity $\Delta \mathbf{v}$ is seen to point directly toward the center of curvature. (See small inset.) Because $\mathbf{a}_{\mathrm{c}}=\Delta \mathbf{v} / \Delta t$, the acceleration is also toward the center; $\mathbf{a}_{c}$ is called centripetal acceleration. (Because $\Delta \theta$ is very small, the arc length $\Delta s$ is equal to the chord length $\Delta r$ for small time differences.)

The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii $r$ and $\Delta s$ are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds $v_{1}=v_{2}=v .$ Using the properties of two similar triangles, we obtain

$\dfrac{\Delta v}{v}=\dfrac{\Delta s}{r}$ 

Acceleration is $\Delta v / \Delta t$, and so we first solve this expression for $\Delta v$ :

$\Delta v=\dfrac{v}{r} \Delta s$

Then we divide this by $\Delta t$, yielding

$\dfrac{\Delta v}{\Delta t}=\dfrac{v}{r} \times \dfrac{\Delta s}{\Delta t}$

Finally, noting that $\Delta v / \Delta t=a_{\mathrm{c}}$ and that $\Delta s / \Delta t=v$, the linear or tangential speed, we see that the magnitude of the centripetal acceleration is 

$a_{\mathrm{c}}=\dfrac{v^{2}}{r},$

which is the acceleration of an object in a circle of radius $r$ at a speed $v .$ So, centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that $a_{\mathrm{c}}$ is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/h than at 50 km/h. A sharp corner has a small radius, so that $a_{\mathrm{c}}$ is greater for tighter turns, as you have probably noticed.

It is also useful to express $a_{\mathrm{c}}$ in terms of angular velocity. Substituting $v=r \omega$ into the above expression, we find $a_{\mathrm{c}}=(r \omega)^{2} / r=r \omega^{2} .$ We can express the magnitude of centripetal acceleration using either of two equations:

$a_{\mathrm{c}}=\dfrac{v^{2}}{r} ; a_{\mathrm{c}}=r \omega^{2}.$

Recall that the direction of $a_{\mathrm{c}}$ is toward the center. You may use whichever expression is more convenient, as illustrated in examples below.

A centrifuge (see Figure 6.9b) 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 $(g)$; maximum centripetal acceleration of several hundred thousand $g$  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.

Figure 6.9 (a) The car following a circular path at constant speed is accelerated perpendicular to its velocity, as shown. The magnitude of this centripetal acceleration is found in Example 6.2. (b) A particle of mass in a centrifuge is rotating at constant angular velocity. It must be accelerated perpendicular to its velocity or it would continue in a straight line. The magnitude of the necessary acceleration is found in Example 6.3.

Of course, a net external force is needed to cause any acceleration, just as Newton proposed in his second law of motion. So a net external force is needed to cause a centripetal acceleration. In Centripetal Force, we will consider the forces involved in circular motion.

Source: Rice University, https://openstax.org/books/college-physics/pages/6-2-centripetal-acceleration#import-auto-id3033074
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