Lecture 1 | Potential Energy & Conservation of Energy

Conservation of Energy

Lecture 1 | Potential Energy & Conservation of Energy

Conservation of Energy

When finding the work done by various types of forces, an interesting result arises from the work done by gravity and elastic springs. The work done by both types of forces only depends on the initial and final positions of the object. This means that we can skip the process of calculating work for gravity and springs, and instead write a function that results in the same value that we would have got if we did calculate the work the traditional way. The functions we define in place of the work due to gravity and springs, are called potential energy functions. This section will help familiarize you with working with potential energy functions. For completeness the potential energy functions that arise from the work due to gravity and the work due to a spring are the following:

Gravitational potential energy ($U^g$): $mgy$, where m = mass, g = gravitational constant, y = height relative to zero

Spring potential energy ($U^S$): $\frac{1}{2}k \Delta x^2$, where k = spring constant, x = distance from equilibrium

If we account for all the interactions an object experiences, we can write a clever summation of specific quantities that will help us describe the motion of the object. The clever expression is known as the conservation of (mechanical) energy. Mathematically it is written as:

$K_i + U^g_i + U^S_i + \sum W_{nc} = K_f + U^g_f + U^S_f$

where K = kinetic energy and U = potential energy. In later sections we will expand on this equation to include more potential energy functions, but for now gravity and spring are the only ones we have came across thus far. Recall energy is a scalar, thus application of conservation of energy will greatly simplify otherwise complicated problems if trying to use kinematics.

Check out the OpenStax trailer to learn about the interplay between kinetic and potential energy.

OpenStax AP Physics Chapter 7: Potential And Kinetic Energy Physics Concept Trailer

Pre-lecture Study Resources

Read the BoxSand Introduction and watch the pre-lecture videos before doing the pre-lecture homework or attending class. If you have time, or would like more preparation, please read the OpenStax textbook and/or try the fundamental examples provided below.

Converation of Energy | Potential Energy and Conservation of Energy

When calculating the work from certain forces, it is discovered that although the work depends on the change in position $\Delta \overrightarrow{r}$ it does not depend on the path that the object took from the initial to final position. We call these conservative forces and for our study we are going to limit ourselves to gravity and elastic springs. This allows us to break the work up into conservative and non-conservative work.

$W = W_{c} + W_{nc}$

Since the conservative work only depends on the change in location - it is path independent - we can define a function of position to account for the work from the conservative forces. An example is the work from gravity (near Earth) is always $W^g = mg \Delta y$, where y is the displacement in the vertical direction from the initial to final location (for more on how this is true see BoxSand's pre-lecture videos below). Since this only depend on the vertical displacement we can account for this work by defining what is called a potential energy function, $U_g=mgy$, and thus the change in potential energy is equal to negative the work.

$\Delta U^g=-W^g=mg \Delta y$

Now we never have to calculate the work from gravity again, we can simply set a new type of initial and final energy, namely the potential energy.

$U^g=mgy$

Now lets make an addendum to the conservation of energy equation.

Recall, $K_i +\Sigma W = K_f$

with, $W = W_{c} + W_{nc}$, and $W_c=- \Delta U^g$

we get, $K_i + (-\Delta U^g) + \Sigma W_{nc} = K_f$

which can be rearranged to yield, $K_i + U_i + \Sigma W_{nc} = K_f + U_f$

Mechanical energy (E) is defined as kinetic plus potential energy (E = K + U), using this notational simplification we arrive at the final form of conservation of (mechanical) energy. This is how you start all energy problems!

$E_i + \Sigma W_{nc} = E_f$

Which says take the initial energy of the system, kinetic plus potential, and add to that the work from non-conservative forces (everything but gravity and elastic springs), and set that equal to the final energy of the system.

If the above explanation is confusing, use the end result and start getting a feel for the mechanics by applying the theorem and using the equations below.

$K = \frac{1}{2}mv^2$

Gravity: $U^g = mgy$

Elastic Spring: $U^s = \frac{1}{2}kx^2$

$E = K + U$

$E_i + \Sigma W_{nc} = E_f$