Imagine a skydiver falling towards the earth. Without their parachute open, the skydiver is falling quickly as their body pushes away the air surrounding them. As they release their parachute, the surface area pushing away the surrounding air is much greater, which causes them to slow down and slightly increases the temperature of the surrounding air. The skydiver makes it safely to the ground, feeling grateful for air resistance! Air resistance is an example of a dissipative force, which in this case is extremely useful! In this article, we will talk about what a dissipative force is and go through an example.
Definition of Dissipative Forces
In the article, "Conservative Forces", we talk about the difference between conservative and non-conservative forces, and we mention a type of non-conservative force: a dissipative force. A dissipative force is a force that decreases the mechanical energy in a system. Dissipative forces acting on an object always oppose the motion of the object, which means they always do negative work. Some examples of this force are the force of friction, air resistance, and fluid resistance.
Dissipative force: a force that decreases the mechanical energy in a system.
Let’s think about what makes friction a dissipative force. Consider a box that is given an initial push and allowed to move across a rough surface, as shown in the image below. The box has kinetic energy due to its motion from the initial push force. The force of friction works negatively on the box, causing the box to slow down and eventually come to a stop. If the surface had been frictionless, the kinetic energy would have remained the same, but the negative work done by the force of friction causes a decrease in the kinetic energy. The kinetic energy is dissipated as heat energy as the box slows down. The heat energy cannot be transformed back into kinetic energy, so there is a decrease in the mechanical energy of the system.
Fig. 1 - The dissipative force of friction slowing down a moving box.
Dissipative Forces and Potential Energy
The conservative forces acting on objects in a system give the system potential energy. Since dissipative forces are all non-conservative forces, the work done by dissipative forces does not contribute to the potential energy. The mechanical energy in a system decreases when dissipative forces are acting on objects in the system.
Difference between Conservative and Dissipative Forces
There are some key differences between what happens when a conservative force acts on objects in a system versus when a dissipative force acts on objects in a system. The table below outlines these differences.
| Conservative Forces | Dissipative Forces |
Contribute to the mechanical energy in a system by giving the system potential energy. | Decrease the mechanical energy in a system. |
Path-independent - the potential energy only depends on the final and starting position of the object. | Path-dependent - the amount of mechanical energy lost depends on the path taken by the object. |
Reversible - the potential energy stored in the system can be converted into kinetic energy later. | Irreversible - some mechanical energy is lost to other forms of dissipated energy, like thermal and sound energy. |
Examples include gravity, the spring force, and the electric force. | Examples include friction, air resistance, and fluid resistance. |
Formulas for Dissipative Forces
The most common dissipative forces to encounter in physics problems are friction and air resistance. The magnitude of the force of friction, \(F_f,\) is found by multiplying the friction coefficient, \(\mu,\) by the normal force \(F_n\):
\[F_f=\mu F_n.\]
The magnitude of the force of air resistance, \(F_\mathrm{air},\) which can also be called the drag force, is proportional to the velocity, \(v,\) squared of the object according to the equation:
\[F_\mathrm{air}=\frac{1}{2} \rho v^2 C_D A,\]
where \(\rho\) is the fluid density, \(C_D\) is the drag coefficient, and \(A\) is the cross-sectional area.
It is important to remember the directions of the force vectors. When finding the net force of a ball falling down, the force of gravity and air resistance point in opposite directions, so a negative sign in front of the force of air resistance would be necessary. It is always helpful to draw a free-body diagram!
Fig. 2 - Air resistance slowing a skydiver's fall to the earth.
As mentioned above, the work done by dissipative forces is always negative, since the forces point in the opposite direction as that of the motion of the object. Including the work done by these forces in the net work done on the system will decrease the kinetic energy, and thus the mechanical energy, in the system.
The equation above for air resistance is an approximation for macroscopic objects. For microscopic objects, the force of air resistance is approximately proportional to the velocity instead of the velocity squared.
Example of Dissipative Forces in a System
It is important to be able to identify which forces are dissipative forces in a system and calculate the work done by these forces. Let’s go through an example to get some practice!
You push a chair \(3\,\mathrm{m}\) horizontally across a rough surface with a push force of magnitude \(F_p=100\,\mathrm{N}.\) The force from friction is \(F_f=50\,\mathrm{N}.\) What is the total work done by the non-conservative forces working on the chair? Determine which of the non-conservative forces are dissipative forces, then find the total work done by them.
The non-conservative forces working on the chair are the friction force and the push force. To find the work done by each non-conservative force, we need to multiply them by the distance traveled and determine if the force is doing positive or negative work on the chair. The work done by the push force is positive because the force vector points in the same direction as the motion of the chair. So the work done by the push force is:
\[\begin{align*}W_p&=F_p d\\[8pt]&=(100\,\mathrm{N})(3\,\mathrm{m})\\[8pt]&=300\,\mathrm{J}.\end{align*}\]
The friction force vector points in the opposite direction compared to the motion of the chair, so it does negative work on the chair:
\[\begin{align*}W_f&=-F_f d\\[8pt]&=-(50\,\mathrm{N})(3\,\mathrm{m})\\[8pt]&=-150\,\mathrm{J}.\end{align*}\]
Taking the sum of these, we get the total work done by the non-conservative forces on the chair:
\[\begin{align*}W_nc&=W_p+W_f\\[8pt]&=300\,\mathrm{J}-150\,\mathrm{J}\\[8pt]&=150\,\mathrm{J}.\end{align*}\]
The dissipative force acting on the chair is the force of friction. We found the work done by the friction force to be \(W=-150\,\mathrm{J}.\)
Dissipative Force - Key takeaways
- A dissipative force is a force that decreases the mechanical energy in a system.
- Dissipative forces acting on an object always go against the motion of the object, which means they do negative work.
- Some examples of dissipative forces are the force of friction, air resistance, and fluid resistance.
- While conservative forces give a system potential energy, dissipative forces decrease the mechanical energy of the system through dissipated forms of energy, such as heat energy.
References
- Fig. 2 - Skydiver (https://pixabay.com/photos/skydiver-parachute-skydiving-flying-2183279/) by GuentherDillingen is licensed by Public Domain.
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