Is tension always the same as centripetal force?

For example, if a ball is attached to a string and released from a vertical height and then pivots around a point to initiate circular motion, tension is equal to centripetal motion.

If, on the other hand, a ball hands from a string and it’s hit in such a way that it travels in a vertical circle. Tension is not just equal to centripetal force.

When is tension equal to centripetal force, and when is it another value?

The scenarios above are taken from previous problems I’ve seen in class. I’m not sure if I’ve explained them as clearly as needed, but I think the general idea should be understood.

• In none of your cases is the centripetal force equal to the tension, you also need to algebraically add the component of the gravitational force away from the center.
– SRS
Oct 16 '19 at 7:48

"Centripetal" is Latin for "towards the center." A centripetal force is not a particular type of force like a frictional force or a magnetic force. It's just a force that makes an object go in a circle. The word "centripetal" describes the direction of the force, not the type of force.

When a car drives around in circles on level ground, the centripetal force is a frictional force. When we whirl a ball around on a string, the centripetal force is a normal force of the string on the hook it's tied to, and the magnitude of this normal force is equal to the tension in the string. (Tension is not a type of force.)

I think that part of the misunderstanding is due to the use of the term centripetal force in a context where more that force is involved.

Take the vertical circle motion as an example.
The circulating mass is under the action of two forces, the gravitational attraction due to the Earth and the tension in a string.
These two forces produce a net force on the mass which causes a centripetal, towards the centre, acceleration.

As the mass progresses it is still acted upon by the same two forces and there is a net force acting on the mass.
That net force can be resolved into a radial component and a tangential component.
The radial component is responsible for the centripetal acceleration of the mass.

Even when only one force acts, as with the Earth in an elliptical orbit around the Sun, one must be careful as the gravitational attraction of the Sun can be split into two with one component causing the centripetal acceleration.
If however the orbit is assumed to be circular then one can say the the gravitational attractive force as a whole causes a centripetal acceleration and thee acceleration is due to a centripetal, towards the centre, force although personally I would avoid calling it that.

"Centripetal force" is only really a thing when something is traveling along a circular arc. For an object that is moving in a circle, the net force required in a direction perpendicular to the instantaneous direction of travel is given this name. If the only force acting on the object is the pull (tension) of a string, then the component of this force that is perpendicular to the motion of the object is the centripetal force. In the case that the string is oriented perpendicular to the object's velocity, the magnitude of the centripetal force is equal to the tension.

If multiple forces are acting on the object (such as, in your vertical example, gravity as well as the pull of a string) then in general the tension force is not equal to the centripetal force, as it is only the net force (the vector sum of all the forces) that accelerates the object. At certain points, however, the tension may still equal the magnitude of the centripetal force, as, for example, when a ball whirling around vertically on a string passes through the horizontal configuration.