6.2.1 Centripetal Force

In the previous section, we have learnt that a body in (uniform) circular motion has a centripetal acceleration of \displaystyle {{a}_{c}}=\frac{{{{v}^{2}}}}{r}=r{{\omega }^{2}}. So to keep a body in circular motion, a centripetal (net) force is needed! Since N2L dictates that {{F}_{{net}}}=ma, the magnitude of the centripetal (net) force must be

\displaystyle {{F}_{c}}=m\frac{{{{v}^{2}}}}{r}=mr{{\omega }^{2}}

So a mass going round and round in a circle requires a (net) force that is always perpendicular to its velocity. As the mass “turns”, the (net) force must also “turn” so that it remains centripetal. As this centripetal force only changes the motion’s direction but not its speed, uniform circular motion is produced.

From energy consideration, it is also clear why the (net) force must be completely centripetal without any tangential component. Recall that W=F\Delta s\cos \theta . A tangential force, being parallel to the velocity, will do some work (either positive or negative) on the mass. This will change the KE and thus the motion’s speed. A centripetal force, being perpendicular to the velocity, does zero work, and therefore allows for the KE to be constant.

Please be very clear that a centripetal force is required to produce circular motion. If an object experiences a centripetal force, it travels along a circular path. If it doesn’t, it doesn’t. It is dangerous (and probably wrong) to think that a circular motion “generates” a centripetal force. Strictly speaking, it is always the force that produces the motion, never the other way round!

Now let’s look at some examples of centripetal forces.

Tension

Consider a ball of mass m tied to a rope of fixed length L swung at speed v. It is the tensional force T in the rope that provides the centripetal force that results in the ball going in a circle of radius L.

\displaystyle T=m\frac{{{{v}^{2}}}}{L}

This comes about naturally because of the ball’s inertia and the rope’s fixed length. All the time, the ball wants to travel along a straight line, but the rope wants to maintain its length L. So the rope will naturally pull the ball just strongly enough to keep the ball at distance L from the centre of the circle. And “just strongly enough” is \displaystyle m\frac{{{{v}^{2}}}}{L}. If the ball is swung too fast and \displaystyle m\frac{{{{v}^{2}}}}{L} is larger than the maximum tension the rope can provide, the rope will simply snap, upon which the ball will switch from circular motion to straight-line motion.

Friction

Consider a merry-go-round spinning at angular velocity ω, where a boy of mass m sits at distance r from the center. The frictional force f exerted by the platform on the boy provides the centripetal force that results in the boy going in a circle of radius r.

f=mr{{\omega }^{2}}

Remember that boy has inertia. On his own, he would rather travel along a straight line. But the platform’s surface underneath his butt is travelling along a circular path. Relative to the platform, his butt is trying to slide radially outward (surprise?). So a radially inward frictional force arises. And the magnitude of the frictional force will naturally be mr{{\omega }^{2}} because that’s what’s needed to “stick” the two surfaces together. In fact, if the platform is spinning so fast that mr{{\omega }^{2}} is larger than the maximum friction the platform’s surface can provide, the boy’s velocity will not change as fast as the surface below him. That’s when the boy starts sliding out of the merry-go-round.

The fictitious “centrifugal force”

The “centrifugal force” is supposed to be a force that is experienced by an object in circular motion. Take for example the gravitron, a popular amusement ride where people in a spinning drum are “pinned” against the wall. In popular science, this is explained as follow: that the spinning motion generates a “centrifugal force” which pushes those people radial outward, thus pinning them against the wall.

But there is a better explanation: that the inertia of those people causes them to want to continue in the tangential direction, whereas the spinning wall wants to turn inward. This causes the people to press into the wall (and thus giving the illusion of a centrifugal force)! The resulting contact force of the wall provides the centripetal force to keep those people in circular motion!

So in H2 physics, it is always a centripetal force (and inertia) that produces circular motion. It is never a circular motion that produces centrifugal force. It is true that the centrifugal force as a fictitious inertia force is a very useful concept if we adopt something called a rotating frame of reference. But since rotating frames of reference is not in the H2 physics syllabus, the centrifugal force is an illegitimate force in H2 examinations.

Demonstration

Ball Bearings in Tabao Container

Gravitron, Wall of Death, and Nuts on Spinning Disc

Circling Nuts on Spinning Disc

Video Explanation

Wall of Death

Animation

F, v and r

Concept Test

Disastrous Turn and Disastrous Spin

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