A racing car runs on an unbanked closed track ABCDEFA made of two concentric circular arcs joined by two straight radial links, all centred at a point O. The outer arc ABC has radius $2R$ and sweeps three-quarters of a circle ($270^\circ$). The inner arc DEF has radius $R$ and sweeps a quarter circle ($90^\circ$). The two arcs are joined by straight radial segments CD and FA, each of length $R$ (the difference $2R-R$ between the two radii). Here $R = 100\ \text{m}$, the coefficient of friction between the tyres and the road is $\mu = 0.1$, and the maximum speed the car can reach on a straight is $50\ \text{m s}^{-1}$. Taking $g = 10\ \text{m s}^{-2}$, find the minimum time to complete one round.

A racing car runs on an unbanked closed track ABCDEFA made of two concentric circular arcs joined by two strai

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Concept understanding — Centripetal Force

Centripetal Force: The Invisible Hand That Keeps Things Going in Circles

Imagine you're in a car taking a sharp turn to the left. You feel yourself being pushed to the right, against the door. That feeling — that's your body trying to keep moving straight while the car turns. Now here's the key insight: you don't actually feel a force pushing you outward. What you feel is your own inertia — your body's natural desire to keep moving in a straight line.

The real force is the one the car door exerts on you, pushing you inward toward the centre of the turn. That inward push is centripetal force.

The Intuition: Why Does Anything Need a Force to Go in a Circle?

Newton's first law says: an object in motion stays in motion in a straight line unless acted on by an external force. A straight line is the "default" path. To make something go in a circle — which is a constantly changing direction — you need a force that continuously pulls it away from that straight line.

Think of a stone tied to a string, whirled around your head. The string is taut. That tension is the centripetal force. If you let go, the stone doesn't fly outward — it flies off tangentially, in a straight line from the point of release. The string was constantly pulling it inward, preventing it from escaping.

Note

The word "centripetal" comes from Latin: centrum (centre) + petere (to seek). It means "centre-seeking." This is the opposite of "centrifugal" (centre-fleeing), which is a fictitious force you feel only in a rotating reference frame — not a real force in physics.

The Precise Statement

Centripetal force is any force that causes an object to follow a curved path, directed toward the centre of curvature of that path. It is not a new, independent force like gravity or friction. It is the name we give to the net force that points radially inward when an object moves in a circle.

For uniform circular motion (constant speed $v$ along a circle of radius $r$ ), the magnitude of centripetal force is:

F_c = \frac{m v^2}{r}

Where:

$m$ = mass of the object
$v$ = speed (magnitude of velocity)
$r$ = radius of the circular path

The corresponding centripetal acceleration (which is always perpendicular to velocity) is:

a_c = \frac{v^2}{r}

This acceleration points toward the centre. It is not constant in direction — it rotates as the object moves — but its magnitude is constant for uniform circular motion.

What Provides the Centripetal Force?

Centripetal force is always supplied by some real physical interaction. Here are common examples:

Situation	What provides centripetal force
Car turning on a flat road	Friction between tyres and road
Satellite orbiting Earth	Gravitational attraction
Stone on a string	Tension in the string
Electron orbiting a nucleus	Electrostatic attraction
A roller coaster looping the loop	Normal force from the track (plus gravity at the top)

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Cornering speed is limited by friction supplying the centripetal force, $v=\sqrt{\mu r g}$ : $10\sqrt2\ \text{m s}^{-1}$ on the wide $2R$ arc and $10\ \text{m s}^{-1}$ on the tight $R$ arc; the straights are run at the top speed $50\ \text{m s}^{-1}$ . Summing time = distance/speed over the four sections gives about $86.3\ \text{s}$ .

Concept: maximum cornering speed

On an unbanked curve of radius $r$ , friction provides the centripetal force: $\dfrac{mv^2}{r}\le\mu mg$ , so the fastest safe speed is

v_{\max}=\sqrt{\mu r g}.

Section speeds ( $\mu=0.1,\ g=10,\ R=100$ )

Outer arc ABC ( $r=2R=200\ \text{m}$ ): $v_1=\sqrt{0.1\times200\times10}=\sqrt{200}=10\sqrt2\approx14.14\ \text{m s}^{-1}$ .
Inner arc DEF ( $r=R=100\ \text{m}$ ): $v_2=\sqrt{0.1\times100\times10}=\sqrt{100}=10\ \text{m s}^{-1}$ .
Straights CD, FA: no cornering limit, so run at the top speed $v_3=50\ \text{m s}^{-1}$ .

Section lengths

ABC $=2R\times\dfrac{3\pi}{2}=3\pi R=300\pi\ \text{m}$ . …

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Concept: maximum cornering speed

Section speeds (μ=0.1, g=10, R=100\mu=0.1,\ g=10,\ R=100μ=0.1, g=10, R=100)

Section lengths

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Section speeds ( $\mu=0.1,\ g=10,\ R=100$ )

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