What Exactly Does Equation (2) Mean? (Equations of Motion from PE function)

In summary, the conversation discusses how to use an equation of potential energy to determine the equations of motion and whether or not solving a partial differential equation is necessary. The potential can be differentiated to find the force vector, but finding the position as a function of time may not always be solvable analytically. This applies to even the case of an inverse square law.
  • #1
humancentered666
1
1
Homework Statement
(There isn't really one, I'm just confused about the interpretation of the equation.)
Relevant Equations
F=ma (1)
Fᵢ({x})=-∂V({x})/∂xᵢ (2)
V({x}) is a potental in a system.
What exactly is this equation telling me? How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE? I'm extremely sorry if this question comes off ignorant.
 
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  • #3
humancentered666 said:
How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE?
If you are given the potential then, as (2) shows, you can differentiate it with respect to position to find the force vector.
If your aim is to find position as a function of time, that is not always solvable analytically. Even the simple case of an inverse square law is nontrivial.
 

FAQ: What Exactly Does Equation (2) Mean? (Equations of Motion from PE function)

1. What is the significance of Equation (2) in the context of equations of motion?

Equation (2) is also known as the equation of motion from the potential energy (PE) function. It is used to describe the relationship between an object's potential energy and its position in a given system. This equation is derived from the principle of energy conservation and helps in understanding the motion of an object in a conservative force field.

2. How is Equation (2) different from other equations of motion?

Unlike other equations of motion, Equation (2) specifically focuses on the potential energy of an object. It takes into account the change in an object's potential energy due to its position in a given system, whereas other equations of motion may consider other factors such as velocity, acceleration, and time.

3. Can Equation (2) be applied to all systems?

Equation (2) can be applied to systems where the forces involved are conservative, meaning that the work done by these forces does not depend on the path taken by the object. Examples of such systems include a simple pendulum, a mass-spring system, and a ball rolling on a smooth surface. It cannot be applied to systems with non-conservative forces, such as friction or air resistance.

4. How is Equation (2) derived?

Equation (2) is derived from the principle of energy conservation, which states that the total energy of a closed system remains constant. It involves calculating the change in potential energy of an object as it moves from one point to another, taking into account the work done by conservative forces. This derivation is based on fundamental concepts of physics, such as work, potential energy, and the principle of energy conservation.

5. What are the applications of Equation (2)?

Equation (2) has various applications in the fields of physics and engineering. It is used to understand the motion of objects in conservative force fields, which is crucial in fields such as mechanics, thermodynamics, and electricity. This equation is also used in designing and analyzing systems such as roller coasters, bridges, and other structures that involve potential energy and motion.

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