homework and exercises - Why doesn't a spinning object in the air fall?

Let's say I have a ball attached to a string and I'm spinning it above my head. If it's going fast enough, it doesn't fall. I know there's centripetal acceleration that's causing the ball to stay in a circle but this doesn't have to do with the force of gravity from what I understand. Shouldn't the object still be falling due to the force of gravity?

Answer

We have the ball orbiting at a distance $R$ from the centre of rotation and the string inclined at angle $\theta$ with respect to the horizontal.

Two main forces act on the ball: gravity $mg$ ($m$ is the mass of the ball, $g$ the Earth's gravitational acceleration) and $F_c$, the centripetal force needed to keep the ball spinning at constant rate. $F_c$ is given by:

$$F_c=\frac{mv^2}{R},$$

where $v$ is the orbital velocity, i.e. the speed of the ball on its circular trajectory.

Trigonometry also tells us that if $T$ is the tension in the string, then:

$$T\cos\theta=F_c.$$

Similarly, as the ball is not moving in the vertical direction, thus $F_{up}$:

$$T\sin\theta=F_{up}=mg.$$

From this relation we can infer:

$$T=\frac{mg}{\sin\theta}.$$

And so:

$$\frac{mg}{\tan\theta}=F_c=\frac{mv^2}{R}.$$

Or:

$$\tan\theta=\frac{gR}{v^2}.$$

From this follows that for small $\tan\theta$ and thus small $\theta$ we need large $v$. But at lower $v$, $\theta$ increases. Also note that $\theta$ is invariant to mass $m$.