13.11 Conservation of energy  (Page 7/7)

However, if we a have a system involving translational energy only, then it allows us to consider rigid body as point mass equivalent to particle of the system. This is a significant simplification as we are not required to consider angular aspect of motion and hence energy associated with angular motion.

System types and conservation law

It may appear that conservation law is subject to system definition. Certainly it is not. We state conservation law in the context of an isolated system for our convenience. We can as well state the law for “open” and “closed” system. Not only that we can have a statement of conservation law considering “universe” as the only system.

Actually, the statement of conservation law as “energy can neither be created nor destroyed”, applies to all systems including universe. For system like “open” or “closed” systems, which allow exchange of energy, we can think in terms of “transfer” of energy. A statement may be phrased like “change in the energy of the system is equal to the energy transferred “to” or “from” the system”.

We can be quite flexible in the application of conservation law with the help of “accounting” concept. We can consider “energy” as “money” in our account. Our account is credited or debited by the amount we deposit or withdraw money. Similarly, the energy of the system increases by the amount of energy supplied to the system and decreases by the amount of energy withdrawn form the system.

Example

Problem 1: An ice cube of 10 cm floats in a partially filled water tank. What is the change in gravitational potential energy (in Joule) when ice completely melts (sp density of ice is 0.9) ?

Ice cube in a tank

Solution :

This question has been included with certain purpose. Though we have not studied “phase change” in the course up to this point, but we can apply our understanding broadly to understand this question. Along the way, we shall point out relevance of this question for the conservation of energy.

Now, gravitational potential energy will change if there is change in the water level or the level of center of mass of ice mass.

The ice cube is 90 % submerged in the water body as its specific density is 0.9. When it melts, the volume of water is 90 % of the volume of ice. Clearly, the melted ice occupies volume equal to the volume of submerged ice. It means that level of water in the tank does not change. Hence, there is no change in potential energy, as far as the water body is concerned.

However, the level of ice body changes after being converted into water. Its center of mass was 4.0 cm below the water level in the beginning, as shown in the figure.

Ice cube in a tank

When ice converts in to water, the center of the converted water body is 4.5 cm below the same water level. Thus, there is a change of level by 0.5 cm. The potential energy of the ice, therefore, decreases :

Ice cube in a tank

$$\Delta U=-mg\Delta h=-V\rho g\Delta h$$

$$\Rightarrow \Delta U=-{0.1}^{3}x0.9X10X0.5=-0.045J$$

We need to account for this energy change. The gravitational energy of the system of “water-ice” can not decrease on its own. We shall come to know that phase change is accompanied by exchange of heat energy. The ice cube absorbs this heat mostly from the water body and a little from surrounding atmosphere. If we neglect energy withdrawn from the atmosphere (only 10 % is exposed), then we can say that energy is transferred from potential energy of “ice-water” system to the internal energy of the system. As such, there is a corresponding increase in the internal energy of the system. This transfer of energy forms take place as “heat” to the ice body. Hence, this is merely a transfer of energy of the system from one form to another.

Here, we do not intend to prove the exactness of change in potential energy with the change in the internal energy in the system. But, the point about accounting of energy, in general, is illustrated by this example.