Why does tension increase with decreasing period?

Question

A student can’t use a yoyo very well, so they swing it around their head in a horizontal circle. They notice the faster they swing it, the more tension there is in the string. They hypothesize that tension increases with decreasing period. Using your knowledge of physics, make a statement that supports their hypothesis. 1. Explain, but do not use an equation. 2. Derive an equation that relates the tension and the period for horizontal swings.

Answer 1

1 .

It would seem that as you are moving the yoyo at the end of the rope, it wouldn't take much force to move an object really slowly and have a large period of rotation.

The yoyo will have a constant mass (some practices call it constant inertia since this term more describes the resistance to movement caused by forces) so for slow and rotating movement, it won't take much force.

If you are to speed it up and have the yoyo make the rotation at a higher rate and thus a lower period, you could infer that because a low force made a high period (a beginning statement of inverse proportionality) that a large force should make a lower period.

2 .

For uniform circular motion:

F_net = m v² / r

v = 2 π r / T

(the rotational speed is equal to the yoyo making it around the circumference of 2 π r in the period T)

Combining the net force equation and the speed equation looks like:

F_net = m ( 2 π r / T )² / r = 4 π² r² m / ( T² r ) F_net = 4 π² m r / T²

The only force acting on the yoyo is the tension so: F_net = F_T

Combining these final two equations results in:

F_T = 4 π² m r / T²

This final equation shows that the tension force, F_T, is inversely proportional to the period squared, T².

So, say you were to quadruple the tension force (multiply it by a factor of 4), the period would be "cut in" half (it would make it around one full rotation in half the time it took the other tension to; half the period).

You could continue to test different periods and see how tension changes or vice versa.

I hope this helps! Message me on the site if you have any more questions concerning this or other problems you may have!