How to show cons. of Energy from F=change of potential

In summary, the conversation discusses the relationship between force and change in potential, and the transformation needed to show the corresponding kinetic energy difference. There is a question about integrating the LHS and where the 1/2 term comes in. The solution involves recognizing that d/dt (r'2) = 2 r' r''.
  • #1
trelek2
88
0
Force is equal to change in potential (r is a point in space suppose integrating between points r1 to r2 in time t1 to t2, i did not indicate this but it is not relevant for my quenstion:
[tex]mr'' =F(r) = -\delta V[/tex]
after multiplying both sides by dr it is obvious RHS will give potential difference, but LHS needs to be transformed to show corresponding kinetic energy difference:
[tex]mr'' \cdot dr = m \int_{}^{} r'' \cdot r' \cdot dt = m \int_{}^{} \frac{d}{dt}r' ^{2}dt [/tex]
:(
The problem is there should be a 1/2 term there as well. I don't know how to integrate this LHS. Can you please help- show me clearly step by stem what is done and where this 1/2 comes in...
 
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  • #2
trelek2 said:
[tex]mr'' \cdot dr = m \int_{}^{} r'' \cdot r' \cdot dt = m \int_{}^{} \frac{d}{dt}r' ^{2}dt [/tex]
:(
The problem is there should be a 1/2 term there as well. I don't know how to integrate this LHS. Can you please help- show me clearly step by stem what is done and where this 1/2 comes in...

Hi trelek2! :smile:

It's because d/dt (r'2) = 2 r' r'' :wink:
 
  • #3
Yes, I am a retard:P
 

FAQ: How to show cons. of Energy from F=change of potential

How can I show the conservation of energy from the change of potential using the formula F=change of potential?

The conservation of energy can be shown by using the formula F=change of potential, where F represents the force and change of potential represents the change in potential energy. This formula states that the force acting on an object is equal to the rate of change of its potential energy. If the potential energy of an object remains constant, then the force acting on it must also remain constant, thus demonstrating the conservation of energy.

What is potential energy and how does it relate to the conservation of energy?

Potential energy is the energy that an object possesses due to its position or configuration. It is a form of stored energy that can be converted into other forms, such as kinetic energy. In the context of the conservation of energy, potential energy is important because it is directly related to the force acting on an object and can be used to show that energy is conserved in a system.

Can the conservation of energy be shown in all systems using the formula F=change of potential?

Yes, the conservation of energy can be shown in all systems using the formula F=change of potential. This formula is a fundamental principle in physics and applies to all types of systems, whether it is a simple pendulum or a complex mechanical system. As long as the potential energy of the system can be calculated and the force acting on it can be determined, this formula can be used to demonstrate the conservation of energy.

How does the change of potential affect the conservation of energy?

The change of potential, or the change in an object's potential energy, is a key factor in demonstrating the conservation of energy. According to the formula F=change of potential, the force acting on an object is directly proportional to the change in its potential energy. Therefore, if the potential energy remains constant, the force must also remain constant, thus illustrating the conservation of energy in the system.

Are there any real-world applications of using the formula F=change of potential to show the conservation of energy?

Yes, there are many real-world applications of using the formula F=change of potential to show the conservation of energy. This formula is commonly used in various fields of science, such as physics, engineering, and mechanics, to analyze and understand the behavior of different systems. It is also used in practical applications, such as calculating the potential energy of objects in motion or determining the work done by a force on an object.

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