Lecture 1 | Work & Kinetic Energy Theorem

Work & Kinetic Energy

Lecture 1 | Work & Kinetic Energy Theorem

Work & Kinetic Energy

Objects can possess energy in many forms but this section is only going to consider the energy from the translational motion of the system. This motional energy is called kinetic energy ($K = \frac{1}{2} m v^2$) and notice it is a scalar quantity. It is true that velocity is a vector quantity but when you square it, all direction is lost. Objects can also exchange energy with other objects through the concept of Work ($W$). The Work-Kinetic Energy theorem relates the change in kinetic energy of the system to the work performed on the system.

Check out this song about all the forms of energy.

Types of Energy Song

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.

Work and Energy| Work and Kinetic Energy Theorem

Consider the motion in 1D and the kinematics equation, $v_{fx}^2 = v_{ix}^2 + 2a_x \Delta x$, and Newton's 2nd Law with constant acceleration in one dimension, $\sum F_x = m a_x$. A simple substitution, and minor rearrangements, of the acceleration from the 2nd law into the kinematics equation yields:

$\frac{1}{2}m v_{fx}^2 - \frac{1}{2}m v_{ix}^2 = \sum F_x \Delta x$

It appears that forces acting over a distance changes the velocity of objects according to the equation above. As it turns out, this manipulation of equations will be of great benefit in analyzing the motion of objects, so we define new terms for the expressions shown in the above equation.

Linear Kinetic Energy (K): $\frac{1}{2}m v^2$.

Work (W): $F_x \Delta x$.

Using our newly defined terms, we now can write what we call, the Work-Kinetic Energy theorem as:

$K_f - K_i = \sum W$

This expression lets you see the heart of the Work-Kinetic Energy theorem, namely that a system's change in energy is due to work from external forces performed on the system. The theorem is often rearranged so that it can be interpreted as: take the initial kinetic energy of the system and add the net work acting on the system and this is equal to the final kinetic energy of the system.

$K_i + \sum W = K_f$

**Note: K and W are scalar quantities! The above arguments are shown in one dimension. Continue reading further in this section to learn how to generalize if our forces, displacements, or velocities are in more than one dimension.