Circular motion and centripetal force

Instantaneous velocity is tangent to the circle, whereas centripetal acceleration points towards its centre. The directions are perpendicular in uniform circular motion. The speed may stay fixed even though the velocity vector changes. An inward acceleration of magnitude $a_c=v^2/r$ follows from the changing direction of velocity. On a force diagram, draw the actual forces first and resolve them towards the centre. A so-called outward centrifugal force is not an additional real force in an inertial frame. If the inward net force disappears, an object moves along its instantaneous tangent rather than continuing around the circle.

Centripetal force is the name for the resultant inward force, not a separate force to add to the diagram. Tyre-road friction can turn a car on a level road, tension can turn a ball attached to a string, and gravity can keep a satellite in orbit. For a level turn with negligible other horizontal forces, static friction must supply $mv^2/r$. If its available magnitude is smaller than this requirement, the assumed circular path cannot be maintained without slipping. Include forces in the radial equation according to their directions, and use a separate equation for vertical equilibrium if needed.

One complete revolution covers a distance $2\pi r$. If the period is $T$, the speed is $v=2\pi r/T$. Substitution gives $a_c=4\pi^2r/T^2$. Be clear about which quantities stay constant in a comparison: at fixed radius, doubling speed requires four times the inward force; at fixed speed, doubling radius halves that force. If a question gives a diameter, halve it before using the radius in a formula. Keep speed in $\mathrm{m/s}$, radius in $\mathrm{m}$ and force in $\mathrm{N}$.