How do you derive this formula?

[Trickster_00]

How do we come to the formula for Conservation of Mechanical Energy?
[Initial Kinetic Energy + Initial Potential Energy = Final Kinetic Energy + Final Potential Energy]

It's not a part of our syllabus, but I'm just curious how to derive that formula.

Also, could you please clarify to me what's the formula for work-energy principle?

My book says it's [Work done = Change in Kinetic Energy]... but what if it involves external forces (e.g., friction) and change in potential energy? I'd appreciate it too if you break down the formula for me. Much appreciated. Thanks.

 

[Feldoh]

You can't really derive conservation of energy it's sort of like what a mathematician would call an axiom.

There are many ways however in which you can see conservation of energy come into play but they all require that it is taken to be true.

 

[wheelatharm]

Feldoh is absolutely right.

When you have work done by an external force, it will either take away or add to the total energy of the system. The energy is not destroyed, just moved from one system to another.

For Friction ---> Initial PE + Initial KE = Final PE + Final KE + Work done by Friction.

M

 

[wheelatharm]

Just to clarify, the work energy principal should state that work done on a system is equal to the total energy change of the system. This means work can increase PE, KE, spring PE, internal energy, etc, of the system.
For example, raising an object slowly and coming to rest has no change on KE, but requires work to increase the PE. Compressing a spring has no effect on KE, but still requires work to increase the spring energy. Even the work done by friction is never lost, it raises the internal energy of the objects in contact.

M

 

[PJC01]

I can provide some insight on how the formula for Conservation of Mechanical Energy is derived. The formula is based on the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the case of mechanical energy, it refers to the energy associated with an object's motion or position.

To derive the formula, we start with the basic definition of mechanical energy, which is the sum of an object's kinetic energy (KE) and potential energy (PE). This can be represented as E = KE + PE.

Now, let's consider a scenario where an object is initially at rest on a surface and then falls under the influence of gravity. At the initial position, the object only has potential energy due to its position above the surface. As it falls, this potential energy is converted into kinetic energy. At the final position, the object has no potential energy but only kinetic energy. This can be represented as:

Initial KE = 0
Initial PE = mgh (where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the surface)
Final KE = 1/2mv^2 (where v is the final velocity of the object)
Final PE = 0

By applying the principle of conservation of energy, we can equate the initial and final mechanical energies:

Initial KE + Initial PE = Final KE + Final PE
0 + mgh = 1/2mv^2 + 0
mgh = 1/2mv^2

This is the formula for Conservation of Mechanical Energy, which states that the initial potential energy of an object is equal to the final kinetic energy of the object.

In regards to your question about the work-energy principle, it is derived from the same concept of conservation of energy. The formula for work done is defined as W = Fd, where F is the force applied and d is the displacement of the object. By substituting the formula for work into the formula for kinetic energy (KE = 1/2mv^2), we get:

W = Fd = 1/2mv^2
Rearranging this equation, we get:

F = (1/2mv^2)/d
which can be simplified to:

F = ma (using the formula for acceleration, a = v^2/2d)

This formula shows that