Question

If you let a mass at the end of a string start swinging, at first the maximum swing decreases rather quickly, but once the swing has become small it takes a long time for further significant decrease to occur. Try it! Explain this simple observation.

Short answer

If you let a mass at the end of a string start swinging, at first the maximum swing decreases rather quickly, but once the swing has become small it takes a long time for further significant decrease to occur. This happens because air resistance is proportional to$v^2$ .

Step-by-step solution

Define the air resistance

Air resistance is a type of friction that happens when air collides with another object (a force that resists motion). It is the force that an item feels as it travels through the air. The two permanent forces of nature that move on any item on Earth are air resistance and gravity.

Explanation for the observation

The main force that opposes the swinging movement of the mass is air resistance $F_{air}$. This force is proportional to $v^2$. So, the total energy of the system (and the maximum swing) will decrease faster when the mass is just starting to swing and the higher speed will be $v$. This decrease rate will diminish through time, because lower speeds can be observed as time passes.

Mathematically: $F_{air}=\frac{1}{2}C\rho A{v}^2$

Therefore, the scenario happens because air resistance is proportional to $v^2$. Thus, as the swinging movement of the mass slows downs, there is a lower decrease rate of energy in the system.