Q170 / 180 · COMEDK 2021 · Physics
For motion under central forces, which quantity will be conserved ?
- AKinetic energy
- Correct answerBMechanical energy
- CPotential energy
- DNone of the above
Solution
A central force is a force that is directed along the line joining the source of the force and the particle, and its magnitude depends only on the distance $r$ from the source. Such a force can be expressed as $\vec{F} = f(r)\hat{r}$.
Since the force is purely radial, the torque $\vec{\tau} = \vec{r} \times \vec{F} = \vec{r} \times f(r)\hat{r} = 0$. Consequently, the angular momentum $\vec{L}$ is conserved.
Furthermore, a central force is a conservative force because it is a function of position only and its curl is zero. For any conservative force field, the total mechanical energy $E = K + U$ is conserved, where $K$ is the kinetic energy and $U$ is the potential energy.
Among the given options, mechanical energy is the quantity that is conserved in motion under central forces.
Answer: Mechanical energy
Since the force is purely radial, the torque $\vec{\tau} = \vec{r} \times \vec{F} = \vec{r} \times f(r)\hat{r} = 0$. Consequently, the angular momentum $\vec{L}$ is conserved.
Furthermore, a central force is a conservative force because it is a function of position only and its curl is zero. For any conservative force field, the total mechanical energy $E = K + U$ is conserved, where $K$ is the kinetic energy and $U$ is the potential energy.
Among the given options, mechanical energy is the quantity that is conserved in motion under central forces.
Answer: Mechanical energy
Hence, the correct answer is option: 'B'.