Energy conservation of a rod in free space

[Volkr16]

Hey everyone,

A meager pea brain (me) is in need of some help from you fine physics gentlemen.
This is a concept I've spend a lot of time pondering on (more than I would like to admit).

Imagine a rod of uniform density in free space. If you apply impulse onto the center of mass then it will gain an energy (transverse) of E=1/2mv^2. If you apply an impulse off the center of mass then it will gain an energy of E=1/2mv^2+1/2Iw^2 (transverse+rotational).

Here is what I think I know: Let's assume that this rod gets hit perpendicular to it's length. No matter where it gets hit, the center of mass will have equal transverse motion for equal impulse applied. In other words Ft=mv for center of mass

So let's assume two scenarios, one where the impulse is applied off the center of mass and one where it is on the center of mass. Make the impulse be so, that the velocity of the center of mass will be the same in both cases (From what i stated before i should be able to use the same impulse anywhere on the rod).

My problem lies in the conservation of energy: I know if the impulse is applied off center i will have rotational energy, but i also know that I will have the same translational energy no matter where the rod get hit (perpendicular on the length). Something isn't right here.

Surely my "what i think i know" must be wrong. Could someone please elaborate why?

Thanks in advance,

Volker

 

[Nugatory]

The same impulse means applying a given force for a given time. Depending on where you apply the force, the distance that the point of application travels during that time will be different - and the energy transferred to the rod is equal to the force times the distance, not the force times the time.

 

[TumblingDice]

A very similar post and thread on this recently:
External Forces on a Rigid Body

 

[Volkr16]

Thanks guys, that cleared it up. I searched a lot before posting this, but didn't find that thread - sorry

 

[PJC01]

Hi Volker,

Great question! It's important to understand that energy conservation is always true, but it can be a bit tricky to understand in certain situations. In this case, let's break down the two scenarios you mentioned and see how energy is conserved in each one.

Scenario 1: Impulse applied at the center of mass

In this scenario, the impulse is applied directly at the center of mass, causing the rod to have only translational energy. As you correctly stated, the energy gained is E=1/2mv^2. This energy is conserved because the impulse is applied directly at the center of mass, which means there is no torque acting on the rod. This means that there is no rotational energy being added to the system, so all of the energy gained is in the form of translational energy.

Scenario 2: Impulse applied off the center of mass

In this scenario, the impulse is applied off the center of mass, causing the rod to have both translational and rotational energy. The translational energy gained is still E=1/2mv^2, but now there is also rotational energy, as you correctly stated, of E=1/2Iw^2. This is where it may seem like energy is not conserved, as there is now more energy present in the system compared to the first scenario. However, energy is still conserved because the impulse applied off the center of mass also causes a torque on the rod. This torque causes the rod to rotate, and as a result, some of the translational energy is converted into rotational energy. The total energy gained in both forms (translational and rotational) is still equal to the impulse applied.

In both scenarios, the total energy gained is equal to the impulse applied. The key difference is in how that energy is distributed between translational and rotational forms. In the first scenario, all of the energy is in the form of translational energy, while in the second scenario, some of the energy is in the form of rotational energy. This is due to the presence of a torque in the second scenario.

I hope this helps clarify the concept of energy conservation in this scenario. Keep pondering and asking questions, it's a great way to deepen your understanding of physics!