Conservation laws and symmetry considerations. Their fundamental importance in physics.
The law of conservation of energy states that energy cannot be created or destroyed and can only be converted from one form to another. A system, such as the room I’m sitting in or the universe, always has the same amount of energy, unless it is added from outside.
This can be illustrated by steam / heat engines, where if you provide heat, you get work and motion. Heat energy is converted to motive energy.
In a gravitational field, as an object falls to earth, its potential energy decreases as height is decreasing and it is converted to kinetic energy as the object gains speed and accelerates in free fall. Kinetic energy is related to movement. Kinetic energy is a result of gaining energy due to movement. Hence to do work, the object has to move. However, potential energy is not a consequence of movement because it is a result of configuration or position as it is a stored energy. It is converted to kinetic energy and vice versa. Due to conversion, as illustrated in a gravitational field, energy is conserved.
Potential energy can also be explained by other means not depending on gravity. Such as winding a spring or an elastic rubber band which can be unwounded to do work and release the band or spring by converting elastic potential energy to kinetic energy.
According to the law of conservation of momentum, in an isolated system, if two or more bodies act upon each other, their total momentum stays the same unless an external force is applied. Hence, momentum, like energy, cannot be created or destroyed.
This law was derived by Newton from his third law of motion. It states that if an object applies a force on another object, the second object applies an equal force on the first object but in the opposite direction. It portrays the famous notion that every action has an equal and opposite reaction.
Consider two colliding particles A and B whose masses are m1 and m2 with initial and final velocities as u1 and v1 of A and u2 and v2 of B. The time of contact between two particles is given as t.
A=m1(v1−u1) (change in momentum of particle A)
B=m2(v2−u2) (change in momentum of particle B)
FBA=−FAB (from third law of motion)
FBA=m2∗a2=m2(v2−u2)t
FAB=m1∗a1=m1(v1−u1)t
m2(v2−u2)t=−m1(v1−u1)t
m1u1+m2u2=m1v1+m2v2
Therefore, above is the equation of law of conservation of momentum where m1u1+m2u2 is the representation of total momentum of particles A and B before the collision and m1v1+m2v2 is the representation of total momentum of particles A and B after the collision.
The follow scenarios portray the law of conservation of momentum perfectly.
In a rocket, initially during takeoff, gas pressure is created and gas is ejected from behind which is the action and causes the rocket to go forward into space, which is the reaction. Newton’s third law of motion also applies here. This example is also applicable in case of jet engines.
A heavy person colliding with a small person has more momentum even if they travel with the same velocity. If both are heavy, masses are more and hence the total momentum is higher than it would be if both people were light and small. If both are heavy they experience more pain than if both were light. Thus total momentum depends on mass more than on velocity. Momentum is important as it gives impact of collision.
Suppose on a boat, I’m sitting and my friend is sitting on another boat.
My mass and boat mass = m. Friend’s and her boat mass = m
If I push her boat forward, my boat goes backward. Third law of motion applies here as an action creates and equal and opposite reaction.
Before the push, total momentum = 0 as both were at rest.
After push, my momentum = mv. Friend momentum = mv. Velocity is same because my push causes her boat to go forward and hence I also moved backward.
Total momentum after the push = mv — mv = 0.
Thus total momentum is conserved in this simple example.
Angular momentum is momentum in circular direction and equals mass X velocity X radius.
If ice skaters spread their hands, they go slow and if they bring their hands together, they go very fast. If radius increases, velocity decreases according to law of conservation of momentum.
I am sitting on a revolving chair with a cycle wheel in hand. If I rotate it, my chair will start moving in opposite direction. When I rotated the wheel, the momentum was in one direction but when it caused my revolving chair to move in opposite direction, the momentum was in the opposite direction. I started to rotate with my chair which means angular momentum is conserved.
Conservation of energy and momentum are like symmetry. If we try to change total energy or momentum, it does not change. Law of conservation of momentum is a consequence of position symmetry. The laws of physics don’t depend on our location implies that momentum is conserved. Law of conservation of angular momentum is a consequence of direction symmetry, which implies that laws of physics don’t depend on which direction in space angular momentum is applied. Law of conservation of energy is a consequence of time symmetry, which implies laws of physics don’t depend on which time we are in.
This connection between conservation laws and symmetry portrays a great beauty in nature. Thus as conservation laws are connected to nature, science and arts collide. This theory was proposed by a notable mathematician contemporary to Einstein, known as Emmy Noether. The fundamental importance of conservation in physics lies here.
More integral qualities of conservation which are pivotal in physics can be illustrated by laws of thermodynamics.
First law of thermodynamics states that total energy in a system never changes. One type can be converted to another form.
In a river dam, when water accumulates, its height increases. Potential energy increases as a result, which is converted to kinetic energy. That is converted to electricity and in houses we get heat and motion.
The second law of thermodynamics states that in any conversion of energy, 100% efficiency cannot be achieved. Total conversion is not possible. As much electricity that we provide, not that much heat we get. Nothing is an ideal environment. Output/input is not equal to 1. Complete conversion is never achieved even if my engine and technology are perfect. This is a pessimistic, inevitable and inviolable law of nature. It is never to be changed. It is a consequence of the fact that energy is dissipated and wasted as heat to the surroundings.
There is a second way of stating it which is heat moves from higher to lower temperature. Heat death is a fate of the universe as it will lose its temperature and so will the sun which will die one day.
Third way of stating it is anything we do leads to chaos and irregularity. Entropy is chaos which never decreases in the universe. After many days of coming back home, room gathers dust as molecules are constantly rearranged. Old never becomes young. Locally entropy may increase when we decorate a room but somewhere else and globally total entropy increases.
The fourth way of stating the law is that we can use it to recognize past from future. It is an indicator of future from past. Entropy is arrow. Wherever entropy increases, that is the future.
The third law of thermodynamics states that absolute zero temperature cannot be achieved ever because it is the theoretical lowest temperature and molecules cannot move any slower.
The contradiction between the first and the second laws is that if total energy is always constant, there should be 100% conversion possible. But the fact that 100% efficiency is not achieved despite total energy in a system remaining constant is a consequence of the fact that energy is dissipated or lost as heat to the surroundings. Energy is not lost from the universe though it is just lost from the system in question. This contradiction is another important feature of conservation and the contradiction makes conservation play an essential role in physics.
More ways in which conservation plays a key role in physics are related here. Conservation and thermodynamics laws are very pessimistic and sad theories. They deal with heat and motion and are important for engines and everything in the universe. Thermodynamics is a completely different branch of physics and deals with heat and motion solely.
If the laws of conservation and thermodynamics were false, we would not be able to explain where extra energy comes from or where the additional energy goes to.
