Centripetal Force

Net inward force toward curvature centers required to maintain curved trajectories.

General idea of the topic

The centripetal force is not a new interaction but the net result of the real forces along the direction pointing toward the centre of the path. A body follows a circle only if that net force has the right value: m·v²/r. It can be the tension of a rope, the friction of a tire, a component of the normal force or gravity, depending on the case.

The topic teaches how to identify which real force plays the centripetal role in each situation, to set up the second law along the radial direction and to analyse the classic cases: flat and banked curves and the vertical loop, where the body must stay in contact with the track.

What the student is expected to understand

By the end, the student should be able to write the radial equation of any circular motion, compute the maximum speed on a curve and the minimum speed at the top of a loop, and explain why the centripetal force does not appear as an extra force on the free-body diagram.

The student should also be able to distinguish between the centripetal force, which exists, and the so-called centrifugal force, which only appears when the motion is described from a rotating frame.

Recommended learning route

1. The Concept of Centripetal Force: defines the centripetal force as the radial net force and relates it to speed and radius.
2. Typical Examples: goes through typical radial examples: a flat curve, a banked curve, a vertical loop and an orbit.
3. Flat and Banked Curves: analyses the speed limit on flat curves, where friction is involved, and the banking that allows the curve to be taken without it.
4. Vertical Loop and Contact Condition: studies the vertical loop, the minimum speed at the top and the contact condition of the normal force.

Physical criterion to acquire

The essential criterion is not to add the centripetal force to the force diagram. Only the real forces are drawn, projected onto the radial axis, and their sum is set equal to m·v²/r. If some real force, such as a normal force, would have to be negative to satisfy the equation, the body loses contact and the circular model no longer holds.

Connection with real problems

• road curves and banked tracks.
• loop rides and centrifuges.
• satellites and approximately circular orbits.
• slings, pendulums and any swing on a rope.

Final goal of the topic

The final goal is to identify the real force that produces a circular motion, set up the radial equation rigorously and analyse the speed limits that guarantee it is maintained.