# The direction of net force in circular motion

The small ball attached by a thin string is in uniform circular motion as shown in the picture (vertical plane). There are two forces acting on the ball, Gravitational force $$F_g$$ and Tension force from the string $$F_T$$, and the direction of these two forces are as shown. I've learned that the direction of acceleration is pointed to the center of the circle, so does the net force. But The net force F$$_{net}$$ I got is shown in the picture, What did I miss?

• The force points to the center when the speed around the circle is constant. That is not the case for vertical circular motion - it is slower on top and faster on the bottom. Sep 14, 2023 at 14:24
• So it will never be uniform vertical circular motion? @MariusLadegårdMeyer Sep 14, 2023 at 14:42

For uniform circular motion, acceleration and force are directed toward the center.

Here you have another component. The ball will slow as it rises on the circular trajectory. On the other side, it will speed up.

As the speed varies, the component of force and acceleration toward the center will vary. $$a_{r}$$ must be $$v^2/r$$ for the path to be circular. As long as the ball rotates fast enough to keep tension in the string, the string will exert enough force to keep the path circular. That is, the tension in the string will adjust itself so that $$F_r = mv^2/r$$.

In short, there is always a radial component of force in circular motion. The total force may or may not be directed toward the center.

• Is there a special case where “always” fails when at the top of the circle, the tension is zero and mg supplies the radial force (for zero time…)?
– JEB
Sep 14, 2023 at 15:18
• @jeb, I don't see how that fails. The statement says there is a net force with at least some component that is radial. In your statement, $mg$ supplies that force. Sep 14, 2023 at 15:47
• @BowlOfRed yes.
– JEB
Sep 14, 2023 at 19:00

I've learned that the direction of acceleration is pointed to the center of the circle, so does the net force. But The net force F$$_{net}$$ I got is shown in the picture, What did I miss?

The net force doesn't point to the center of the circle. The centripetal force responsible for the centripetal acceleration does. Only in the case of uniform (constant speed) circular motion does the net force point to the center.

For non uniform circular motion (which this is) the centripetal force is one of the components of the net force. But there is also a tangential force responsible for the change in speed (tangential acceleration) of the ball. That is the other component of the net force. Both components of the net force are changing in time.

For a complete analysis of how to determine the speed at various positions, and thus the centripetal force, see: http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/cirvert.html

Hope this helps.