Circular motion and centripetal acceleration

Constant speed does not mean zero acceleration

Acceleration is the rate of change of velocity. Since velocity includes direction, an object can accelerate even when its speed is constant. Uniform circular motion is the clearest example. A car moving around a circular track at constant speed has a velocity vector that continuously changes direction.

The acceleration points toward the center of the circle and is called centripetal acceleration:

a_c=rac{v^2}{r}

where $v$ is speed and $r$ is radius.

Direction of centripetal acceleration

At any instant, velocity is tangent to the circular path. Centripetal acceleration points inward, perpendicular to the velocity. It changes the direction of velocity, not its magnitude, in uniform circular motion.

If the inward acceleration disappeared, the object would move in a straight line tangent to the circle, according to Newton's first law.

Angular description

Circular motion can also be described with angular variables. Angular speed is

omega=rac{d heta}{dt}

For uniform circular motion,

$v=romega$

Substituting into $a_c=v^2/r$ gives

$a_c=romega^2$

The period $T$ is the time for one revolution, and frequency $f$ is revolutions per second:

omega=rac{2pi}{T}=2pi f

Centripetal force

Centripetal force is not a new kind of force. It is the net inward force required to produce centripetal acceleration. Newton's second law gives

F_{net,inward}=mrac{v^2}{r}

The inward force may be tension, gravity, friction, a normal force, or a combination of forces.

For a ball on a string moving in a horizontal circle, tension provides the inward force. For a car turning on a flat road, static friction provides the inward force. For a satellite in orbit, gravity provides the inward force.

Car turning on a flat road

A car of mass $m$ turns on a flat circular path of radius $r$ at speed $v$. The required inward force is

mrac{v^2}{r}

If static friction supplies this force, then

f_s=mrac{v^2}{r}

Since $f_sleq mu_sN$ and $N=mg$, the maximum safe speed satisfies

mrac{v^2}{r}leq mu_smg

so

$vleq sqrt{mu_sgr}$

This shows why higher friction and larger radius allow safer turns at higher speeds.

Vertical circular motion

In vertical circular motion, gravity changes direction relative to the center as the object moves. At the top of a loop, gravity may point toward the center. At the bottom, gravity points away from the center while the normal force or tension points upward toward the center.

The radial equation must be written carefully at each point:

sum F_r=mrac{v^2}{r}

where inward is chosen as the positive radial direction.

Misconception: centrifugal force

In an inertial frame, there is no outward force required for circular motion. The real net force points inward. The feeling of being thrown outward in a turning car comes from your body's inertia: it tends to continue in a straight line while the car turns inward. In a rotating non-inertial frame, an apparent centrifugal force can be introduced, but it is not a real interaction force.

The big idea

Circular motion requires inward acceleration because velocity direction changes. The required net inward force is $mv^2/r$. Centripetal force is not a separate force type; it is the role played by actual forces such as tension, gravity, friction, or normal force. Circular motion problems are solved by identifying the radial direction and applying Newton's second law inward.