Conservative Force - Classical Mechanics - Problems And Solutions

Physics Problems And Solutions - Conservative force, in physics, any force, such as the gravitational force between the Earth and another mass, whose work is determined only by the final displacement of the object acted upon. 

The total work done by a conservative force is independent of the path resulting in a given displacement and is equal to zero when the path is a closed loop. Stored energy, or potential energy, can be defined only for conservative forces.

Nonconservative forces, such as friction, that depend on other factors, such as velocity, are dissipative, and no potential energy can be defined for them.

Central force:

1. It is a force whose magnitude depends only on the distance between the object and the origin.
2. It is a conservative field, can be expressed as $F =-\bigtriangledown V$  (the negative gradient of a potential energy).
3. Gravitational force, Coulomb force, and Elastic Force (Harmonic Oscillator) are examples of central (conservative) forces.
4. In conservative field, the net work done by the force is zero, $W =\oint _c F.dr = 0$ ; the total mechanical energy is conserved.
5. Conservative force is irrotional (torque = 0), since curl $\small \bigtriangledown V \textrm{ or } \bigtriangledown \times \bigtriangledown V=0$ .
6. Torque, $\tau = \frac{dL}{dT} = 0$ ; angular momentum, L is conserved
7. Central force = centripetal force (F_g = F_c) produces circular orbit.

Suppose that the gravitational force law between two massive objects were

\[F_{12} = \frac{r_{12} Gm_1.m_2}{r_{12}\! ^{ \, \, (2+\epsilon )}}\]

where ɛ is a small positive number. Which of the following statements would be which is not true?

1. The total mechanical energy of the planet-Sun system would be conserved.
2. The angular momentum of a single planet moving about the Sun would be conserved.
3. The periods of planets in circular orbits would be proportional to the $\left (\frac{3+\epsilon }{2} \right )$ power of their respective orbital radii.
4. A single planet could move in a stationary non circular elliptical orbit about the Sun.
5. A single planet could move in a stationary circular orbit about the Sun.

Solution:

1. TRUE.Gravitational force is a conservative force.
   In conservative field, the total mechanical energy is conserved.
2. TRUE
   In conservative field, angular momentum, L is conserved.
3. TRUE
   \[F_g = F_c\\ \frac{GMm}{r^{(2+\epsilon )}} = mr\omega ^2\\ \frac{GMm}{r^{(2+\epsilon )}} = mr\left (\frac{2\pi}{T} \right )^2\\ \frac{GM}{r^{(3+\epsilon )}} = 4\left (\frac{\pi}{T} \right )^2\\ T^2 = \frac{4\pi^2r^{(3+\epsilon )}}{GM}\\\\ T \propto r^{(\frac{3+\epsilon }{2} ) }\]
4. FALSE
   Central force = centripetal force (F_g = F_c) produces circular orbit. Non central forces do not produce circular orbit.
5. TRUE

Answer: D