# What is rotational potential energy?

I don't understand the concept of rotational potential energy.

During linear motion, when a force is applied, the work it does gets converted to kinetic energy and there is no change in the potential energy. Similarly, during rotational motion when a torque is applied to angularly accelerate a body, the work done by the torque leads to an increase in kinetic energy. Hence the total energy of the body also increases. This video says that the total energy of the body remains constant, hence there is a need for rotational potential energy in order to balance out the increase in the kinetic energy.

Shouldn't the total energy of the "system" including the person applying torque be conserved and the increase in kinetic energy be justified on the basis of the person's energy being decreased? Where does potential energy come into the picture?

• "During linear motion, when a force is applied, the work it does gets converted to kinetic energy and there is no change in the potential energy." Is this really true, though? Where does the force come from? If the force is due to gravity or electrostatics, then it can be thought of as the negative gradient (directional derivative) of the potential. I.e. left alone, the system will feel forces that tend to push it to lower potential energy. – Jabavu Adams Nov 22 '20 at 1:06
• If the force is due to some external agent, that can certainly also change the potential energy. For instance, lifting an object against gravity, or pushing an object against electrostatic repulsion both increase the potential energy of the whole system. – Jabavu Adams Nov 22 '20 at 1:08

The key to understanding the (rather poorly narrated) video is that the lecturer implies (at T=2:30) that $\Delta E=0$ from which it follows that $\Delta KE= - \Delta PE$
• I know it's four years since you posted this, but, well, this is when I'm finally learning this stuff in school. The same way that the other variables and relations have analogues in rotational dynamics to their translational counterparts, would $\Delta PE=m\cdot a\cdot\Delta s$ correlate with $\Delta PE_{\theta}=I\cdot\alpha\cdot r\cdot\theta$? – DonielF Jul 16 '20 at 23:37
• @DonielF not sure of your notation but if the first terms on the right hand side indicate torque, then I think the answer is yes. Because in the first equation $ma=F$, and change in work is of course $F\cdot ds$ (again, I have to assume things about your notation... so while what you wrote looks directionally correct I cannot be sure unless you define your variables). – Floris Jul 16 '20 at 23:43
• @Floris I accidentally threw in an extra $r$ but yes, the idea'd be torque time displacement. $I=mr^2$ is moment of inertia, $\alpha$ is rotational acceleration, $r\theta$ is displacement. So it'd be $PE = Fs = mar\theta = mr^2\alpha\theta = I\alpha\theta$, where $F$ and $a$ are tangential force and acceleration. – DonielF Jul 17 '20 at 0:03