Question

Find the relation between external force and potential energy stored.

Answer 1

Explanation:

Potential Energy from Force: definition

For a conservative force (defined here), we define the potential energy as: U(x) = - \int_{x_{ref}}^{x} F(x')\; dx'

The choice of reference point does now matter because when observable quantities are calculated using the potential energy, it is only the difference between the potential at two different points (e.g. initial and final as in the equation above) that has observable effects. Another view of the zero point is that the potential at a particular point is defined as an indefinite integral whose upper limit is that point. Since integration is uncertain by an additive constant, and the reference point basically determines that constant.

In this course, we will mostly deal with the following conservative forces. It is helpful to do the integrals in advance and have the form of the potential energy ready to use in problems. The following table is a summary of these forces and their potential energies:

Interaction Force Potential Energy

Gravitational, near earth \displaystyle F= mg \displaystyle U(h)= mgh where h is the height

Gravitational, Universal F= \displaystyle -\frac{G m_1 m_2}{r^2} U(r)= \displaystyle -\frac{G m_1 m_2}{r} Where r is the distant between two objects

Spring \displaystyle F= -kx \displaystyle U(x)= \frac{1}{2}kx^2 where x is the distance from the equilibrium point of the spring

Force from Potential Energy

What if you know the potential energy, but not the force in terms of which it was derived? Given that the potential energy is negative the integral of the force, it should be clear that

F(x) = -\frac{dU}{dx}

i.e. the force is the negative of the derivative of the potential energy with respect to position. This means that if the potential decreases with increasing x, then the force is in the positive x direction. This makes sense: as the particle moves to the right, its potential energy will decrease - therefore, if energy is conserved, it's kinetic energy will increase which can only happen if the force is in the direction of motion.

If you think of a graph of the potential energy (note elastic potential energy below) vs. position as the height above ground of a frictionless track (e.g. of a roller coaster), then it's clear that an upward sloping track will push the particle to the left (due to the normal force).

ElasPot.png

A particle to the right of the origin feels a force back toward the origin - i.e. a restoring force proportional to the negative of the displacement.

Turning Points, Equilibria

Given a potential energy curve \displaystyle U(x) (e.g. the one above) you can determine several important things about the motion of a single particle with total energy \displaystyle E_{total}.

Importantly, you also know the force on the particle at any point - it is determined by \displaystyle F_x=-\frac{dU}{dx}.

Answer 2

Newton’s second law of motion is closely related to Newton’s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton’s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. Before we can write down Newton’s second law as a simple equation giving the exact relationship of force, mass, and acceleration, we need to sharpen some ideas that have already been mentioned.

sorry for the last answer okkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk