Centripetal Force

Use the centripetal-force method whenever an object follows a circular path, or one well approximated as circular, and you need the inward force that keeps it curving. Centripetal force is the inward part of the net force; if it disappears, the object stops curving and flies off along a tangent. The steps below keep you from treating it as a mysterious extra force.

Step 1: Identify The Circular Path

Confirm the motion follows a path of radius $r$, because the whole relation depends on that radius. For motion at speed $v$ on a circle of radius $r$, the required inward force magnitude is

This tells you how much inward force is needed, not which physical force supplies it. You may also see it as

since $v = \omega r$ for circular motion.

Step 2: Use The Inward-Force Formula

For speed $v$ on the path, the inward component of the net force must satisfy $F_c = m v^2 / r$. The reason the force points inward is that a moving object tends to keep going straight; to keep bending its path into a circle, the net force has to keep aiming at the center. Even at constant speed the velocity changes, because its direction keeps turning, and that directional change requires centripetal acceleration

Step 3: Find The Physical Source

Decide which real force provides that inward component: tension, friction, gravity, or a normal force. This is the actual physics step. The formula gives the size of the required inward force, but you still have to name the real force that delivers it. A ball on a string is the classic case: the string tension pulls toward the center, so the velocity keeps turning, and if the string breaks there is no new outward force, the ball simply continues straight.

Step 4: Check Whether Speed Is Changing

If the speed changes, there is also a tangential component of acceleration, so the total net force is not purely centripetal. In that case $m v^2 / r$ still gives the inward part of the net force, but not the whole net force.

Full Worked Example: Car On A Flat Curve

A car of mass $m = 1200\ \mathrm{kg}$ moves around a flat circular curve of radius $r = 50\ \mathrm{m}$ at speed $v = 15\ \mathrm{m/s}$.

Step 1: circular path, radius $50\ \mathrm{m}$. Step 2: apply the inward-force formula,

So the car needs $5400\ \mathrm{N}$ of net force toward the center. Step 3: on a flat road, that inward force is usually provided by static friction between the tires and the road, the real physics step. Step 4: at constant speed there is no tangential component here, so the net force is purely centripetal.

Where Each Step Trips People Up

Step 1 (identify the path): Ignoring the radius. For the same mass and speed, a smaller radius means a sharper turn and a larger centripetal force. Self-check: what is $r$, and did I use it?

Step 2 (use the formula): Calling circular motion balanced. If the path keeps curving, the net force is not zero; a nonzero inward force is required. Self-check: is there a net inward force, or did I set forces to zero?

Step 3 (find the source): Treating centripetal force as a separate force, or inventing an outward force in an inertial frame. Ask which real force points inward; the "outward" feeling in a turning car comes from your body tending to keep moving straight. Self-check: which actual force supplies the inward pull?

Step 4 (speed changing): Forgetting the tangential component when speed varies. Self-check: is the speed constant, or is there also tangential acceleration?

Where The Method Is Used

Centripetal force appears whenever motion follows a circular or near-circular path: cars turning, objects on strings, roller-coaster loops, satellites, and planets in orbit. In many problems the main job is to connect circular motion to another force law, setting tension equal to $m v^2 / r$, or gravity equal to $m v^2 / r$, depending on the situation.

To build intuition, keep the same car and radius but double the speed from $15\ \mathrm{m/s}$ to $30\ \mathrm{m/s}$. Because the force depends on $v^2$, the required centripetal force becomes four times larger, not two, and you can recheck which real force would have to provide that pull.