Unit 8: Potential Energy and Conservation of Energy

8.1 - Potential Energy

Conservative Forces

A force is a conservative force if the net work it does on a particle moving around any closed path, from an initial point and then back to that point, is zero. Equivalently, a force is conservative if the net work it does on a particle moving between two points does not depend on the path taken by the particle.

The gravitational force and the spring force are examples of conservative forces; while the kinetic frictional force is a non-conservative force.

Potential Energy

Potential energy is energy that is associated with the configuration of a system in which a conservative force acts.

When the conservative force does work $W$ on a particle within the system, the change $\Delta U$ in the potential energy of the system is:

\Delta U = -W

As a practical example of this, when an object is lifted up with work $W$ , the resulting change in its potential energy $\Delta U$ is equal to the inverse of $W$ .

If a particle moves from point $x_i$ to $x_f$ , the change in potential energy can be calculated using an integral:

\Delta U = -\int_{x_i}^{x_f} F(x) \space dx

Gravitational Potential Energy

The potential energy associated with a system consisting of Earth and a nearby particle if gravitational potential energy.

If the particle moves from height $y_i$ to height $y_f$ , the change in gravitational potential energy is:

\Delta U = mg \space \Delta y

Elastic Potential Energy

Elastic potential energy is the energy associated with the state of compresssion or extension of an elastic object (spring). For a spring that exerts a spring force $F = -kx$ when its free end has displacement $x$ , the elastic potential energy is:

U(x) = \frac{1}{2} k x^2

In this reference configuration, the spring is at its relaxed length where $x = 0$ and $U = 0$ .

8.2 - Conservation of Mechanical Energy

The mechanical energy $E_{mec}$ of a system is the sum of its kinetic energy $K$ and potential energy $U$ :

E_{mec} = K + U

In an isolated system with no external forces causing energy changes (no friction, only conservative forces), them the mechanical energy $E_{mec}$ of the system cannot change. This is known as the conservation of mechanical energy, and can be written as:

K_2 + U_2 = K_1 + U_1

in which the subscripts refer to different moments in time within the system. This principle can also be written as:

\Delta E_{mec} = \Delta K + \Delta U = 0

8.3 - Reading a Potential Energy Curve

If we know the potential energy function $U(x)$ for a systemin which a one-dimensional force $F(x)$ acts on a particle, we can find the force as:

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

If $U(x)$ is given on a graph, then at any value of $x$ , the force $F(x)$ is the inverse of the slope of the curve at that point, and the kinetic energy of the particle is given by:

K(x) = E_{mec} - U(x)

where $E_{mec}$ is the total mechanical energy in the system.

A "turning point" is a point $x$ at which the particle reverses its motion (where $K = 0$ ).

A particle is at equilibrium at points where the slope of the $U(x)$ curve is zero (where $F(x) = 0$ ).

8.4 - Work Done on a System by an External Force

Important Ideas

Work $W$ is the energy transferred to or from a system by means of an external force acting on the system.

When more than one force acts on a system, their net work is the transferred energy.

When friction is not involved, the work done on the system and the change $\Delta E_{mec}$ in the mechanical energy of the system are equal:
$W = \Delta E_{mec} = \Delta K + \Delta U$

When a kinetic frictional force acts within the system, then the thermal energy $E_{th}$ of the system changes. Therefore, the work done on the system is then:
$W = \Delta E_{mec} + \Delta E_{th}$
The change $\Delta E_{th}$ is related to the magnitude $F_k$ of the frictional force and the magnitude $d$ of the displacement caused by the external force:
$\Delta E_{th} = F_k d$