Law of Conservation of Energy - Equations and Examples
Conservation refers to the condition where there is no change. So, The variable in an equation, which represents a conserved quantity, is constant over time. Its value remains constant both before and after a particular event.
There are many such conserved quantities in physics. They help in making predictions and making a lot of complicated situations much easier. The three fundamental quantities which are conserved in mechanics are energy, momentum, and angular momentum.
Though energy changes, it still remains conservative in nature. This is because while referring to energy, it is the total energy of a system that is considered. The movement of objects and external factors change one form of energy into another but essentially, energy is not lost anywhere. But, it should be noted that the theory of conservation of energy should only be applied to isolated systems.
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This is why a ball rolling across a rough floor is considered not to obey the law of conservation of energy because it is not isolated from the floor. The floor is doing work on the ball through friction, causing changes in the whole system.
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However, the ball and floor together form an isolated system and thereby the law of conservation can be applied and it has been proved that it holds true in the combined ball-floor system.
What do we Mean by a System?
The system can be considered as the suffix we give to a collection of objects. These objects are modeled by the many standard equations. When we describe the motion of an object using the theory of conservation of energy, then the system is supposed to include the object of interest and all other objects that it interacts with.
But, it should be noted that in practice, some of these interactions have to be ignored and omitted in order to solve problems or make predictions. The system we define has objects and interactions that are of importance to us. We generally concentrate only on them and only they are included. 'Environment' is the term used to call the things we don't include. While this creates an overall inaccuracy, it is considered negligible. One of the qualities that a good physicist should possess is the common sense and ability required to differentiate what should be included in the environment.
For example,
When we consider a problem involving a person bungee jumping from a bridge. At the very minimum, the system should consist of the jumper, bungee, and the Earth. If a more precise calculation should be done, one should include the air, which does work on the jumper via drag, or air resistance. To increase the accuracy, one can include the bridge and its foundation, but the bridge is obviously much heavier than the jumper, hence it can be included in the 'environment'.
What is Mechanical Energy?
Mechanical energy can be described as the sum of the potential energy and kinetic energy in a system.
Only conservative forces are associated with mechanical energy. This includes forces like gravity and the spring force. Potential energy is associated with this kind of forces. Nonconservative forces like friction and drag do not fall in this category. With conservation forces, if they are added to a system, the energy imperatively retrieved. But on the other hand, recovering energy of the nonconservative forces is very difficult. This is because it often ends up as heat or some other form which mostly ends up outside the system. This can be also described as an energy being lost to the environment.
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In terms of problems and calculations, this means that mechanical energy is much easier to be calculated and used to make predictions. But, it should be remembered that the conservation of mechanical energy only applies when all forces are conservative. Fortunately, in most cases, nonconservative forces can be ignored and included as part of the environment. A good approximation can be made even without adding up the non-conservative forces.
How can Conservation of Energy Explain the Movement of Objects?
When energy is conserved, it is possible to set up equations since we can equate the sum of the different forms of energy in a system. This allows us to solve the equations for velocity, distance, or any other parameter on which the energy depends. Another advantage of using the theorem of conservation of energy to solve problems is that even if we don’t know all the necessary variables to solve a certain type of problem, it might still be useful in understanding a situation, even in terms of variables.
In a problem which discusses the case of a golfer on the moon striking a golf ball wherein the ball leaves the club at an angle of 45 degrees to the lunar surface traveling at 20 m/s both vertically and horizontally at a total velocity of 28.28 m/s. The question of how high the ball would go can be solved by using the equations associated with the law of conservation of energy and mechanical energy. Such equations are known as kinematic equations.
5 kinematic equations
1: S = Vt
2: \[V_{f} = V_{i} + at\]
3: \[V_{av} = \frac{V_{f} + V_{i}}{2}\]
4: \[S = V_{i}t + \frac{1}{2} at^{2}\]
5: \[2as = V_{f}^{2} - V_{i}^{2}\]
Why can Perpetual Motion Machines Never Work?
The perpetual motion discusses the concept of a machine which continues its motion forever with a condition that there won’t be any reduction in speed. It is the dream of modern science. Though it seems really interesting such a machine cannot really work according to physics we have explored. In fact, even if such a machine were to exist, it wouldn’t be useful to anyone is what has been discovered .apparently, it would have no ability to do work.
According to the principles of mechanics, a system, if it can be fully isolated from the environment and made to subject to only conservative forces, then energy would be conserved and it would run perpetually. The problem we would face is that that in reality, there is no way to completely isolate a system. Also, energy can never be completely conserved within the machine.
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Modern science has made extremely low friction flywheels. They rotate in a vacuum in order to store energy. Yet, we have seen that they still lose energy and eventually spin down when unloaded, over a period of years which can be predicted if other factors of the machine and environment are taken into consideration and calculated accordingly. Many physicists and researchers have thought of the earth itself as an extreme example of such a machine. While it rotates on its axis, interactions with the moon, tidal friction, and other celestial bodies, it too is gradually slowing. An unknown but very interesting and terrifying fact is, every couple of years, scientists add a leap second to our record of time to account for the variation in the length of day.
FAQs on Law of Conservation of Energy
1. What is the Law of Conservation of Energy?
The Law of Conservation of Energy states that for an isolated system, the total energy remains constant over time. This fundamental principle means that energy cannot be created or destroyed, but it can be transformed from one form to another, such as from potential energy to kinetic energy, or transferred from one object to another within the system.
2. Can you provide a real-world example of the Law of Conservation of Energy in action?
A simple example is a swinging pendulum. At the highest point of its swing, the pendulum bob momentarily stops and has maximum potential energy and zero kinetic energy. As it swings downwards, this potential energy is converted into kinetic energy. At the lowest point of the swing, its speed is maximum, meaning it has maximum kinetic energy and minimum potential energy. This transformation continues back and forth, and if we ignore air resistance and friction, the total mechanical energy (potential + kinetic) remains constant throughout the swing.
3. How is the conservation of mechanical energy represented by a formula?
When only conservative forces (like gravity) are acting on a system, the total mechanical energy is conserved. This is expressed by the formula:
E_initial = E_final
or
(KE_i + PE_i) = (KE_f + PE_f)
Where KE is the kinetic energy (½mv²) and PE is the potential energy (mgh). This equation shows that the sum of initial kinetic and potential energies is equal to the sum of the final kinetic and potential energies.
4. Does the Law of Conservation of Energy apply in all situations?
The law itself is universal, but its application depends on how you define the 'system'. The conservation of mechanical energy specifically holds true only in an ideal, isolated system where there are no non-conservative forces like friction or air resistance. In most real-world scenarios, these forces are present and convert some mechanical energy into heat, sound, or other forms that leave the system. For example, a ball rolling on a rough surface slows down because friction converts its kinetic energy into heat. While the ball's mechanical energy is lost, the total energy of the ball-floor-air system is still conserved.
5. If energy is always conserved, why do we need to conserve energy resources?
This is a crucial distinction between the scientific law and the societal practice. The Law of Conservation of Energy states that the total amount of energy in the universe is constant. However, energy can be converted into forms that are not useful for doing work, such as low-grade thermal energy (heat) that dissipates into the environment. When we talk about 'conserving energy resources' like electricity or fuel, we mean preserving these high-quality, concentrated forms of energy and using them efficiently, rather than allowing them to degrade into less usable forms.
6. Why can't a perpetual motion machine exist according to the Law of Conservation of Energy?
A perpetual motion machine is a hypothetical device that could do work indefinitely without an energy source. This violates the Law of Conservation of Energy for two key reasons. Firstly, such a machine would have to create energy from nothing to do work, which is impossible. Secondly, even if a machine were designed to just run forever without doing work, it's impossible to completely eliminate non-conservative forces like friction and air resistance in reality. These forces would continuously convert some of the machine's mechanical energy into heat, causing it to eventually slow down and stop.
7. What is the difference between conservative and non-conservative forces?
The key difference lies in how they affect a system's mechanical energy:
- Conservative Forces: These are forces, like gravity or the elastic force in a spring, where the work done is independent of the path taken. The energy used to work against a conservative force is stored as potential energy and is fully recoverable. For example, the energy you use to lift a book is stored as gravitational potential energy and can be fully converted back to kinetic energy if you drop it.
- Non-Conservative Forces: These are forces, like friction or air drag, where the work done depends on the path taken. The energy lost to these forces is dissipated from the system, usually as heat, and cannot be easily recovered as mechanical energy. The longer the path, the more energy is lost.
