Animated Solution for Physics - Work, Energy, and Power: A particle is moving in a circular path of radius $a$ under the action of an attractive potential $U=-\frac{k}{2r^2}$. Its total energy is

The problem of a particle moving under the influence of an attractive potential is a classic and beautiful application of classical mechanics. It elegantly ties together the concepts of conservative force fields, circular motion dynamics, and the conservation of mechanical energy. Let's embark on a detailed journey to understand the physics behind this scenario.

Imagine a particle of mass $m$ gracefully tracing a circular path of radius $a$. This motion isn't happening by chance; the particle is bound by an invisible tether—an attractive potential field. The potential energy of the particle in this field is given by the function $U(r)=-\frac{k}{2r^2}$, where $k$ is a positive constant and $r$ is the radial distance from the center of the force.

The negative sign in the potential energy is crucial. It signifies that the potential is attractive, meaning the particle is in a "potential well." To move the particle infinitely far away (where potential energy is typically defined as zero), we would have to do positive work against this attractive field.

Force from Potential

To understand the dynamics of the particle, we first need to know the exact force acting on it. In physics, there is a profound and fundamental relationship between a conservative force and its associated potential energy: the force is the negative spatial gradient of the potential energy.

For a purely radial potential, this relationship simplifies to a single derivative:

This equation is our master key. It tells us that the force pushes the particle in the direction where potential energy decreases most rapidly. Let's apply this to our specific potential function.

We need to differentiate our potential energy function, $U(r)=-\frac{k}{2r^2}$, with respect to $r$. Let's rewrite the function slightly to make the differentiation more straightforward:

Now, applying the power rule for differentiation:

The result is a force that depends on the inverse cube of the distance. Notice the negative sign that remains. This confirms our earlier intuition: the force is directed inwards, towards the origin $r=0$. It is an attractive central force.

Since the particle is moving in a circular path of a specific radius $a$, the magnitude of the force acting on it at any point on this path is:

Now, let's shift our focus from the cause of the force to its effect. The particle is executing uniform circular motion. Newton's laws of motion dictate that for any object to move in a circle, there must be a net force acting on it directed towards the center of the circle. This is the famous centripetal force.

The magnitude of the required centripetal force for a particle of mass $m$ moving with speed $v$ in a circle of radius $a$ is given by:

In our scenario, what is providing this centripetal force? It is precisely the attractive central force we just calculated! Therefore, we can equate the magnitude of our attractive force to the required centripetal force.

By equating the two force expressions, we set up the crucial dynamic balance of the system:

Our goal is to find the total energy, which requires knowing the kinetic energy. The kinetic energy $K$ is defined as $\frac{1}{2}m{v}^2$. Let's manipulate our force balance equation to isolate the kinetic energy term.

First, multiply both sides by $a$:

Now, simply divide by 2:

We have successfully found the kinetic energy of the particle! Notice that it is a positive quantity, as kinetic energy must always be.

Total Mechanical Energy

The total mechanical energy $E$ of a conservative system is the sum of its kinetic energy $K$ and its potential energy $U$.

We know the kinetic energy is $K=\frac{k}{2a^2}$. We also know the potential energy at the specific radius $r=a$ is $U(a)=-\frac{k}{2a^2}$.

Let's substitute these values into our total energy equation:

The kinetic energy and the potential energy are exactly equal in magnitude but opposite in sign. When added together, they perfectly cancel each other out, resulting in a total mechanical energy of zero.

Finding that the total energy is zero is not just a mathematical curiosity; it carries deep physical meaning.

In bound systems (like planets orbiting the sun or electrons orbiting a nucleus), the total energy is negative ($E<0$). This means the particle is trapped in the potential well and cannot escape to infinity. Conversely, if the total energy is positive ($E>0$), the particle has more than enough energy to overcome the attractive forces and will escape to infinity, never to return.

A total energy of exactly zero ($E=0$) represents the boundary condition. It means the particle has precisely the minimum amount of energy required to escape the potential well. If it were to break free from its circular orbit, it would travel out to an infinite distance, and its speed would asymptotically approach zero as it got there. It is the threshold between being bound and being free.

This elegant problem demonstrates how the specific mathematical form of a potential energy function dictates the delicate balance of kinetic and potential energies required for stable circular motion.