How can tangential acceleration from a radial force be explained?

Question

A mass is attached to a rope, and put into a circular motion. If I pull the string from the center, the tangential speed of the mass will increase (by conservation of angular momentum). I am applying a force only in the radial direction, so how can the tangential velocity increase if there is no tangential force?

Answer 1

This is a good question and it's fun to think about. You start with a mass in steady circular motion: the force F is going to be mv²/r. This is the force required to keep the mass circling at the same radius. Now suppose you double this force: the mass is going to curve more sharply, it will begin to move in a circle with radius r/2 tangent to the original circle with radius r. This new path brings the mass closer to the center of the circle. So now the motion of the mass and the force applied are no longer perpendicular. Some component of the force is now acting parallel to the motion of the mass: this will do work on the mass, and accelerate it to a higher speed.

Answer 2

Are we looking at an impulsive force in pulling the string from the center?

Then neither momentum or energy is conserved in the process for the system being the particle in rotation.

We would look at the velocity vector for the particle moving in the first circle and the new velocity vector for the particle moving in the inner circle as a function of the pull.

The motion of the object would be that of a spiral arc from the original orbit to the new orbit.

Imagine two concentric circles and a spiral arc that is tangent to both circles as we take snap shots in time.

What do we see, if we do this experiment…the object takes on a spiral trajectory where both the tangential and perpendicular velocity are changing.

We see this example for an electron moving through a constant magnetic field where the Electric charges on a plate capacitor are varied.

These spirals have been studied as ideals:

https://www.mathcurve.com/courbes2d.gb/archimede/archimede.shtml