Final answer:
The energy required to split a liquid drop into 64 smaller drops, in relation to its surface tension, can be calculated by understanding that the total surface area increases with the number of smaller droplets created.
Step-by-step explanation:
If a student asks about the energy needed to break a liquid drop of radius r into 64 smaller drops, they are inquiring about a concept in physics related to surface tension. Surface tension is a property of the surface of a liquid allowing it to resist an external force, due to the cohesive nature of its molecules. The surface tension, t, is the energy required to increase the surface area of a liquid.
The formula for the energy, E, needed to split a single liquid drop into several smaller drops can be derived considering that the total surface area of the smaller drops combined is greater than that of the original single drop. When a drop of radius r is split into 64 drops of radius r/4 (since the volume remains constant), the surface area increases, and thus, more energy is required to overcome the surface tension.
As a result, considering that the surface area of a sphere is given by 4πr², and the number of droplets formed is 64, the change in surface area is proportional to the size of the new droplets, leading to the conclusion that the energy needed would be 8πr²t, where t is the surface tension of the liquid.