Virial Theorem Calculator
Virial Theorem Calculator computes kinetic and potential energies for a particle in a potential, verifying the virial theorem.
Formulas Used in Virial Theorem Calculator
The calculator applies the virial theorem for a single particle:
Virial Theorem:
\\[ \langle T \rangle = -\frac{1}{2} \sum_i \langle \mathbf{r}_i \cdot \mathbf{F}_i \rangle \\]Harmonic Potential:
\\[ V = \frac{1}{2} k r^2, \quad \mathbf{F} = -k \mathbf{r}, \quad \langle T \rangle = \langle V \rangle = \frac{3}{2} k_B T \\]Gravitational Potential:
\\[ V = -\frac{G M m}{r}, \quad \mathbf{F} = -\frac{G M m}{r^2} \hat{r}, \quad \langle T \rangle = -\frac{1}{2} \langle V \rangle \\]Where:
- \\(\langle T \rangle\\): Average kinetic energy (J)
- \\(\langle V \rangle\\): Average potential energy (J)
- \\(k_B\\): Boltzmann constant (\\(1.380649 \times 10^{-23} \, \text{J/K}\\))
- \\(T\\): Temperature (K)
- \\(k\\): Spring constant (N/m)
- \\(G\\): Gravitational constant (\\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\\))
- \\(M\\): Central mass (kg)
- \\(m\\): Particle mass (kg)
- \\(r\\): Distance (m)
Example Calculation
Example: Harmonic Potential, \\(k = 1 \, \text{N/m}, T = 300 \, \text{K}\\)
\\[
\langle T \rangle = \frac{3}{2} k_B T = \frac{3}{2} \times 1.380649 \times 10^{-23} \times 300 \approx 6.212921 \times 10^{-21} \, \text{J}
\\]
\\[
\langle V \rangle = \langle T \rangle \approx 6.212921 \times 10^{-21} \, \text{J}
\\]
Result: \\(\langle T \rangle \approx 6.212921 \times 10^{-21} \, \text{J}, \langle V \rangle \approx 6.212921 \times 10^{-21} \, \text{J}\\).