We can associate a potential energy only to certain type of forces $F_{c}$. We call these forces **conservative forces**.

$$\int\limits_{r_{1}}^{r_{2}} \vec{F}_{c}.d\vec{s} = W_{c} = U_{1}-U_{2} =-\Delta U$$

Gravitational force and the force that the spring exerts are both conservative forces. The energy that is “stored” as potential energy can be converted back to kinetic energy.

Some forces are non-conservative. E.g. Friction and air resistance. They are dissipative forces whereby mechanical energy are reduced when acted upon by these forces.

Note that not all non-conservative forces are dissipative.

### Properties Of Conservative Forces

A force is conservative if the work it does on a particle moving between any two points is **independent** of the path taken by the particle.

An equivalent statement of the above: Work done by a conservative force exerted on a particle moving through any closed path is zero. (I.e. If the particle’s **initial and final point is the same** and the only force acting on the particle is conservative, the work done on the particle by the conservative force is zero.)

### Non-conservative Force

Suppose an object given an initial velocity $v_{i}$ slides on a rough horizontal surface for a distance $d$ before reaching a final velocity $v_{f}$:

$$\begin{aligned} \Delta K &= \frac{1}{2} mv_{f}^{2}-\frac{1}{2} mv_{i}^{2} \\ &=-f_{k}d \end{aligned}$$

Loss in kinetic energy is $f_{k}d$, which is the energy dissipated by the force of kinetic friction. Path of the energy is transferred to the internal energy of the block, and another part to the rough horizontal surface.

### Law Of Conservation Of Energy (With Non-conservative Forces)

Friction and other non-conservative forces remove/add mechanical energy from/to the system. This is the amount by which the mechanical energy of the system changes, denoted by $W_{\text{other}}$:

$$K_{i} + U_{i} + W_{\text{other}} = K_{f}+U_{f}$$

In the case of kinetic friction and air resistance, since the force and displacement are always in opposite direction, the work done is always negative, and the final mechanical energy is always less than the initial mechanical energy (where d is the distance travelled):

$$K_{i} + U_{i}-f_{k}d=K_{f}+U_{f}$$

Experimentally, we observed that change in internal energy is:

$$\Delta U_{\text{internal}} =-W_{\text{other}}$$

Hence, we can write the law of conservation of energy as:

$$K_{1}+U_{1}-\Delta U_{\text{int}} = K_{2}+U_{2}$$

OR

$$\Delta K + \Delta U + \Delta U_{\text{int}} = 0$$

From the above, we can see that:

- Energy lost by non-conservative forces (e.g. friction) can be converted into internal energy in the body (rise in temperature) associated with atomic vibration. These internal atomic motion has kinetic energy and potential energy associated with it.
- Energy can never be created or destroyed.
- Energy may be transformed from one form to another, but the total energy of an isolated system is always constant.

### Potential Energy Function

In general, since work done by a conservative force is a function only of a particle’s initial and final coordinates, we can define a potential energy function $U$:

$$U \left( \vec{R} \right) =-\int\limits_{\vec{r}_{0}}^{\vec{r}} \vec{F}.d\vec{s} + U_{0}$$

Or in 1-D:

$$U \left( x \right) =-\int\limits_{x_{0}}^{x} F_{x} \, dx + U_{0}$$

$U_{0}$ is often taken to be zero at some arbitrary (convenient) reference point. Only the change in potential energy is physically significant.

Conservative force in 1-D:

$$\int F \, dx =-\Delta U$$

In differential form:

$$dU =-F \, dx$$

The conservative force is related to the potential energy function by:

$$F =-\frac{dU}{dx}$$

A conservative force acting on an object within a system equals the negative derivative of the potential energy of the system with respect to x.

The above equation is not new to you. Take a look at the equation for a spring:

$$\begin{aligned} F_{\text{spring}} &=-\frac{dU_{\text{spring}}}{dx} \\ &=-\frac{d}{dx} \left( \frac{1}{2} kx^{2} \right) \\ &=-kx \end{aligned}$$

You will obtain the relation between the force and extension of a spring, which is as expected.

Another example is Gravitational force:

$$\begin{aligned} F_{g} &=-\frac{dU_{g}}{dr} \\ &=-\frac{d}{dr} \left( -\frac{GMm}{r} \right) \\ &=-\frac{GMm}{r^{2}} \end{aligned}$$

### Energy Diagrams & Equilibrium

The motion of a system can be understood qualitatively through its potential energy curve.

There are three types of equilibrium when the force on the object is zero.

Positions of **stable equilibrium** correspond to those points for which $U \left( x \right)$ has a minimum value.

Positions of **unstable equilibrium** correspond to those points for which $U \left( x \right)$ has a maximum value.

Positions of **neutral equilibrium** correspond to a region where $U \left( x \right)$ is constant.

### Problem-Solving Strategy For Conservation Of Energy

- Define your system (fields, springs, other sources of potential energy) and choose initial and final points
- Select a reference position for zero potential energy and write expressions for potential energy associated with each force.
- If mechanical energy is constant then solve by using $E_{i} = E_{f}$ or $K_{i} + U_{i} = K_{f} + U_{f}$
- If friction or air resistance (non-conservative forces) is present, mechanical energy is not constant. Use: $E_{i} + W_{\text{other}} = E_{f}$

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