degrees of freedom mechanics with examples

Degree of freedom

Definition: – The minimum number of independent variables or coordinates required for specifying the position of a dynamical system consisting of one or more particle is called Degree of freedom.

For the $N$ number of particles moving freely in $d$ dimensional space degrees of freedom is represented by the following equation.


Where $N$ is the number of particles and $d$ denote the dimensions of the particle.

If there are constraints then


where $k$ is the number of constraints.

Degrees of freedom Examples

Example 1 :-

Consider a particle moving along x-direction in three dimensional space then in this case

$d = 3$, $N = 1$ and $k = 2$

Constraint $k = 2$ means that particle cannot move in y and z directions. So in this case degrees of freedom would be $f=1$.

Again the degree of freedom would be $f=1$ when the particle is moving in the x-direction in one-dimensional space.

Example 2:-

Let us consider another case where two particles are moving freely in XY plane. Here in this case

d = 2 \hfill \\
N = 2 \hfill \\
k = 0 \hfill \\
\end{gathered} $

here $k=0$ because particles are not constrained to move in XY-plane.

Now degrees of freedom is defined as $f=Nd-k$ which gives us $f=4$