IGNOU MPH 6 2026 - Determine the radial and the angular frequencies and explain their sig

Question

Determine the radial and the angular frequencies and explain their significance.

19 Jan 2026

Answer :

Word Count : 346

For a particle in circular motion or in a central force field, the radial frequency (\omega_r) and angular frequency (\omega_\theta) are defined as follows: Consider a particle of mass (m) moving under a central potential (V(r)). The Lagrangian in polar coordinates ((r, \theta)) is: [ L = \frac{1}{2} m \left(\dot{r}^2 + r^2 \dot{\theta}^2\right) - V(r) ] The radial motion equation is obtained from the Euler-Lagrange equation: [ m \ddot{r} = m r \dot{\theta}^2 - \frac{dV}{dr} ] Angular momentum is conserved: [ L_z = m r^2 \dot{\theta} = \text{constant} = \ell ] (\dot{\theta} = \frac{\ell}{m r^2}) Substitute into radial equation: [ m \ddot{r} = \frac{\ell^2}{m r^3} - \frac{dV}{dr} ] Define effective potential: [ V_{\text{eff}}(r) = V(r) + \frac{\ell^2}{2 m r^2} ] Then

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