High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Physics Textbook (Master) > Work and Energy

Conservative and Non-Conservative Forces

Work-Energy TheoremPrevious Section
Conservation of Energy Next Section
Popular Courses
Find My Course
0:00 / 0:00

Conservative vs Non-Conservative Forces - Part 1


There are two types of work coming from different forces: conservative and nonconservative (nc).
  • Conservative forces perform work that do not depend on the path taken (gravity, spring force). The energy is converted into potential energy.
  • Nonconservative forces perform work that depend on the path (friction). The energy is dissipated as heat or sound. Any non-zero nonconservative work will change the mechanical energy.

Wize Tip
Work done by conservative forces is path independent and work done by non-conservative forces is path dependent.

Conservative Forces and Potential Energy

Work done by any conservative force is proportional to the change in the potential energy associated with that force:

Wc=−ΔU\boxed{W_{c}=-\Delta U}

  • Potential energy is a type of energy which is stored inside the object and can be converted into other energies such as kinetic energy.
  • Potential energy is usually shown by UU, PEPEor UpU_p
  • Gravitational Potential Energy (Earth): ΔUg=mgΔy\Delta{U_g} = mg\Delta{y}
  • Gravitational Potential Energy (Any Planet): ΔUg=−Gm1m2r2\Delta{U_g} = -G\frac{m_1m_2}{r^2}
  • Spring Potential Energy: Us=12kx2U_s = \frac{1}{2}kx^2

  • A conservative force is always toward the points with lower potential energies.
  • If work is done by only conservative forces, then the mechanical energy of the system is conserved.

  • Potential energy can also be described as the negative integral of the dot product of conservative forces and distance as a mathematical expression:
ΔU=−∫r1r2Fc⃗.dr⃗\boxed{\Delta{U} = -\int_{r_1}^{r_2}\vec{F_c}.\vec{dr}}
  • As a differential equation the above equation can be written as:
Fc⃗=−dUpdr⃗\boxed{\vec{F_c}=-\frac{dU_p}{\vec{dr}}}

0:00 / 0:00

Conservative vs Non-Conservative Forces - Part 2


Example: Gravitational Potential Energy

Work done by gravity pull can be written in terms of Gravitational potential energy and is calculated by:


Wgravity=−ΔUg=−(mghf−mghi)W_{gravity}=-\Delta U_g=-\left(mgh_f-mgh_i\right)

Ug=mghU_g=mgh
where h is the height and is measured relative to the reference potential energy.

  • We can also express potential energy due to gravity on any planet by:
Ug=Gm1m2rU_g = \frac{Gm_1m_2}{r}


Watch Out!
There is no well-defined reference point for potential energies. You can pick any point as the zero point of energy and measure potential energy of all other points based on that. That is why we usually talk about change in potential energy instead of potential energy itself!


Example: Elastic Energy of Springs

Similarly, the restoring force of a spring is a conservative force and the potential energy stored inside a spring is given by:
FS=−kΔx                  US=12kΔx2F_S=-k\Delta x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ U_S=\frac{1}{2}k\Delta x^2

Which of the following forces are non-conservative? (select all that apply)


a. Drag
b. Friction
c. Gravity
d, Electrostatic Force
e. Magnetic Force
f. Spring Force


a, b
Which of the following forces are conservative? (select all that apply)