I Energy of spinning objects as axis of rotation moves

AI Thread Summary
When an object like a throwing knife spins without a fixed axis, its axis of rotation naturally converges to its center of mass to minimize its moment of inertia (I). The energy of rotation remains constant if no resistive forces act on it, leading to an increase in angular velocity as I decreases. The relationship between energy, moment of inertia, and angular velocity is described by the equation E = (1/2) I ω², where both I and ω are time-dependent. The total kinetic energy of the object combines translational and rotational components, represented by T = (M/2) v² + (1/2) ω Θ ω, where M is mass and Θ is the inertia tensor. Understanding these dynamics is crucial for analyzing the motion of spinning objects in physics.
Trollfaz
Messages
143
Reaction score
14
Imagine an object, e.g throwing knife, spins in the air but not forced to rotate about a particular axis, i.e no rod impaling it and forcing it to spin about the rod. Then the axis of rotation converges to it's center of mass (CM) to minimize I. But there's nowhere for it's rotation energy to go assuming no resistive forces of the medium.
$$E=\frac{1}{2} I \omega^2= k$$
Both I and ##\omega## are functions of t and
##\frac{dI}{dt}<0##
So angular velocity increases?
$$\frac{dE}{dt}=\frac{dI}{dt}\omega^2+ 2I\omega\frac{d\omega}{dt}=0$$
$$\frac{d\omega}{dt}=-\frac{dI}{dt}\omega^2/2I\omega>0$$
And if we know the rate of change of I we can solve this differential equation to find how angular velocity evolves
 
Physics news on Phys.org
Your expression for the energy only describes the rotational energy of the body around the given axis but not the translational energy of the body as a whole. It's most simple to choose the center of mass as the body-fixed reference point. Then the total kinetic energy of the rigid body reads
$$E_{\text{kin}}=T=\frac{M}{2} \dot{\vec{R}}^2 + \frac{1}{2} \vec{\omega} \hat{\Theta} \vec{\omega},$$
where ##M## is the total mass and ##\hat{\Theta}## is the tensor of inertia around the center of mass of the body.
 
Hello everyone, Consider the problem in which a car is told to travel at 30 km/h for L kilometers and then at 60 km/h for another L kilometers. Next, you are asked to determine the average speed. My question is: although we know that the average speed in this case is the harmonic mean of the two speeds, is it also possible to state that the average speed over this 2L-kilometer stretch can be obtained as a weighted average of the two speeds? Best regards, DaTario
This has been discussed many times on PF, and will likely come up again, so the video might come handy. Previous threads: https://www.physicsforums.com/threads/is-a-treadmill-incline-just-a-marketing-gimmick.937725/ https://www.physicsforums.com/threads/work-done-running-on-an-inclined-treadmill.927825/ https://www.physicsforums.com/threads/how-do-we-calculate-the-energy-we-used-to-do-something.1052162/
Back
Top