Energy and its Conservation

Introduction to Potential Energy

Potential energy ($U$) is the energy stored in a system because of the position or configuration of its parts. Kinetic energy is the energy of motion. Potential energy is stored energy that can later be converted into kinetic energy or used to do work. Potential energy is associated only with a specific class of forces called conservative forces. We will define them formally in the next section. For now, the two most common examples are gravity and the elastic force of a spring.

Suppose a system moves from an initial configuration to a final one. The change in potential energy ($\Delta U$) is defined as the negative of the work done ($W_c$) by the conservative force:

$$\qquad \Delta U = U_f - U_i = -W_c$$

This definition is crucial. It means:

• When a conservative force does positive work (e.g. gravity pulling an object down), the potential energy of the system decreases.
• When a conservative force does negative work (e.g. an external agent lifting an object against gravity), the potential energy of the system increases.

Kinetic energy has a natural zero point: zero speed. Potential energy is defined only by its *change*. We are free to choose a convenient reference configuration and define the potential energy there to be zero ($U = 0$). The choice of this zero point is arbitrary and does not affect the physics. Only the *difference* in potential energy matters.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its vertical position in a gravitational field. The position is measured from a chosen reference level. Consider an object of mass $m$ near the Earth's surface. There the gravitational force is approximately constant, $F_g = mg$, acting downwards.

Lift the object vertically from an initial height $y_i$ to a final height $y_f$, through a height $h = y_f - y_i$. The work done by the gravitational force is:

$$\qquad W_g = F_g d \cos(180^\circ) = (mg)(h)(-1) = -mgh$$

The work is negative because the force of gravity is opposite to the direction of displacement. Now, using the definition of potential energy change:

$$\qquad \Delta U_g = U_f - U_i = -W_g = -(-mgh) = mgh$$

Choose the reference level $y_i = 0$, where $U_i = 0$. Then the gravitational potential energy $U_g$ at any height $h$ above this level is:

$$\qquad U_g = mgh$$

Where you measure from is your choice

A height is only ever a height above something. Nothing in the physics picks that something for you. Choose the floor and a book on a table has one potential energy. Choose the table top and the same book has another.

Elastic Potential Energy

Elastic potential energy is the energy stored in an elastic object, such as a spring. It is stored when the object is stretched or compressed from its equilibrium (natural length) position. The force exerted *by* the spring is given by Hooke's Law, $F_s = -kx$, where $x$ is the displacement from equilibrium.

Now stretch the spring from an initial displacement $x_i$ to a final displacement $x_f$. The work done by the spring force is:

$$\qquad W_s = \int_{x_i}^{x_f} F_s(x) dx = \int_{x_i}^{x_f} (-kx) dx = -\left[ \frac{1}{2}kx^2 \right]_{x_i}^{x_f} = -(\frac{1}{2}kx_f^2 - \frac{1}{2}kx_i^2)$$

The change in elastic potential energy is the negative of this work:

$$\qquad \Delta U_s = U_f - U_i = -W_s = -\left[-(\frac{1}{2}kx_f^2 - \frac{1}{2}kx_i^2)\right] = \frac{1}{2}kx_f^2 - \frac{1}{2}kx_i^2$$

Choose the equilibrium position ($x_i = 0$) as the reference point, where the potential energy is zero ($U_i = 0$). Then the elastic potential energy $U_s$ stored in the spring at a displacement $x$ is:

$$\qquad U_s = \frac{1}{2}kx^2$$

This formula holds whether the spring is stretched ($x > 0$) or compressed ($x < 0$). The stored energy depends on $x^2$.

Whose energy is it?

We say "the ball's potential energy", and it is a convenient shorthand for something slightly different. The energy is stored in the separation between the ball and the Earth. It took both of them to store it.

Introduction

In our discussion of potential energy, we introduced the idea of conservative forces. Now we will define them more formally. Forces in nature can be divided into two categories: conservative and non-conservative. This distinction is fundamental to the principle of conservation of energy.

Conservative Forces

A force is considered conservative if the work it does on an object moving between two points is independent of the path taken. It depends only on the initial and final positions.

There is an equivalent definition. A force is conservative if the work it does on an object moving through any closed path is zero. A closed path is one where the starting and ending points are the same.

Properties of a Conservative Force:

1. Work done is path-independent.
2. Work done over a closed loop is zero ($\oint \vec{F}_c \cdot d\vec{r} = 0$).
3. It can be associated with a potential energy function ($\vec{F}_c = -\nabla U$).

The relationship $\Delta U = -W_c$ that we defined earlier is a direct consequence of a force being conservative. The work done depends only on the endpoints. So the change in potential energy, $U_f - U_i$, has one unique value between those two points.

Examples of Conservative Forces

• Gravitational Force: The work done by gravity when you lift a book depends only on the change in height. It does not matter whether you lift it straight up or take a winding path.
• Elastic Spring Force: The work done by a spring when it is stretched or compressed depends only on its initial and final displacements from equilibrium.
• Electrostatic Force: The work done by the electric field on a charge is also path-independent.

Why the test matters at all

The test is not a curiosity. Suppose the work between two points depends on the route. Then no number attached to a *point* can have a change equal to that work. A potential energy is exactly such a number.

So everything in the next four nodes rests on passing this one test: $U(x)$, energy diagrams, conservation of mechanical energy. A force that fails it does not get a potential energy. Its work has to be carried around explicitly in every calculation.

Non-Conservative Forces

A force is non-conservative if the work it does on an object depends on the path taken between the initial and final points. The work done by a non-conservative force over a closed loop is generally not zero.

Non-conservative forces typically involve dissipative processes like friction or air resistance. These processes convert mechanical energy into other forms, such as heat or sound. Because energy is removed from the mechanical system, you cannot define a potential energy function for these forces.

Examples of Non-Conservative Forces

• Kinetic Friction: Consider pushing a box across a rough floor from point A to point B and then back to A. The friction force always opposes the motion. On the way to B, friction does negative work. On the way back to A, it *also* does negative work. The total work over the closed path is a negative value, not zero. Furthermore, if you take a longer, roundabout path from A to B, you will do more work against friction.
• Air Resistance (Drag): Like friction, the drag force always opposes motion. So the work it does depends on the length of the path traveled.
• Tension in a Rope: While tension can do work, it is generally non-conservative. The work done by tension depends on the specifics of the path and how the tension changes.
• Applied Forces (Pushes/Pulls): A push or pull from an external agent is typically non-conservative.

Where the energy actually goes

Nothing is destroyed when friction acts. The mechanical energy, kinetic plus potential, falls. An equal quantity of heat appears in the block and in the floor.

The whole classification, on one page

Every force this course uses falls on one side or the other of the test. The two questions, does the path matter, and is there a potential energy, always give the same answer.

What are Central Forces?

A central force $\vec{F}$ is one that is always directed along the line connecting a particle to a fixed point. That point is called the center of force. Central forces are a special and important class of forces in physics. Furthermore, the magnitude of the force depends only on the distance $r$ from this center, i.e. $F = F(r)$.

Mathematically, a central force can be written as:

$$\vec{F} = F(r)\hat{r}$$

where $\hat{r}$ is a unit vector pointing from the center of force towards the particle. The most prominent examples of central forces are the gravitational force between two masses and the electrostatic force between two charges. Both follow an inverse-square law.

Central Forces are Conservative

All central forces are conservative. We prove this by showing that the work done by a central force between two points is independent of the path taken.

Consider a particle moving from an initial point $A$ to a final point $B$ under the influence of a central force. The infinitesimal work done is $dW = \vec{F} \cdot d\vec{s}$, where $d\vec{s}$ is the infinitesimal displacement vector. In polar coordinates, this displacement can be written as the sum of a radial component and a tangential component: $d\vec{s} = dr\hat{r} + r d\theta\hat{\theta}$.

We now compute the dot product for the work:

$$dW = (F(r)\hat{r}) \cdot (dr\hat{r} + r d\theta\hat{\theta})$$

Since $\hat{r}$ and $\hat{\theta}$ are orthogonal unit vectors, $\hat{r} \cdot \hat{r} = 1$ and $\hat{r} \cdot \hat{\theta} = 0$. The expression simplifies:

$$dW = F(r) dr$$

To find the total work done from A to B, we integrate this expression:

$$W_{A \to B} = \int_{r_A}^{r_B} F(r) dr$$

This result is important. It shows that the work done by a central force depends only on the initial and final radial distances ($r_A$ and $r_B$). It does not depend on the particular path taken between them. Since the work is path-independent, the force is, by definition, conservative. This is why we can define a potential energy function for forces like gravity and the electrostatic force.

Every path is arcs and radial steps

The argument generalises in one move. Approximate any route by a staircase of arcs at constant radius and steps straight in or out. The arcs cost nothing. The radial steps between the same two radii add to the same total whichever route you took.

Introduction

The conservation of energy is one of the most fundamental principles in physics. By combining the Work-Energy Theorem with the idea of conservative forces, we can derive a powerful tool for analyzing motion.

Choosing a System

First, we must define our system. The choice of system is crucial as it determines which forces are internal and which are external. Energy is conserved within a closed, isolated system. In such a system, no external force does work and no internal non-conservative force is present.

Total Mechanical Energy

The total mechanical energy ($E$) of a system is the sum of its kinetic energy ($K$) and its total potential energy ($U$). The potential energy $U$ is the sum of all types of potential energy within the system (e.g. gravitational, elastic).

$$\qquad E = K + U$$

The Generalized Work-Energy Principle

Consider the net work done on an object, split into work done by conservative forces ($W_c$) and work done by non-conservative forces ($W_{nc}$):

$$\qquad W_{net} = W_c + W_{nc}$$

From the Work-Energy Theorem, $W_{net} = \Delta K$. From the definition of potential energy, $W_c = -\Delta U$. Substituting these gives:

$$\qquad \Delta K = -\Delta U + W_{nc}$$

Rearranging this yields the generalized work-energy principle:

$$\qquad \Delta K + \Delta U = W_{nc} \quad \text{or} \quad \Delta E = W_{nc}$$

This is a more complete statement about energy. It says that the change in the total mechanical energy of a system is equal to the work done by all non-conservative forces. If $W_{nc}$ is negative (e.g. from friction), the system's total mechanical energy decreases.

The Principle of Conservation of Mechanical Energy

A special and very important case arises when the work done by non-conservative forces is zero ($W_{nc} = 0$). This happens if there are no non-conservative forces, like friction or air resistance. It also happens if they act but do no work, for example a normal force perpendicular to the motion.

In this case, $\Delta E = 0$. This means the total mechanical energy of the system remains constant.

$$\qquad E_f = E_i \quad \text{or} \quad K_f + U_f = K_i + U_i$$

This is the Principle of Conservation of Mechanical Energy. It is a powerful problem-solving tool. It relates the speeds and positions of an object at two different points without analyzing the motion in between. The only condition is that conservative forces alone are doing work.

The same total, instant by instant

A falling ball is the smallest example that shows the whole idea. The potential bar shrinks and the kinetic bar grows. Their sum is the same number at every height.

Why the shape of the track drops out

The most useful thing about conservation is what it does not need to know. It relates two instants and says nothing about the route between them. So a complicated frictionless track is no harder than a simple one.

And when it fails

Put friction on the same ramp and the total bar is shorter at the bottom than it was at the top. Mechanical energy is not conserved. Total energy still is, and the difference is sitting in the surfaces as heat.

Draw the boundary before you do the bookkeeping

Gravity may appear as an external force doing work, or as a potential energy stored inside the system. The choice is not a fact about gravity. It is a consequence of where you drew the system boundary.

Worked Examples