Energy of spinning objects as axis of rotation moves

In summary, the energy of spinning objects is closely linked to their axis of rotation. As the axis moves, the distribution of mass relative to the axis changes, affecting the object's moment of inertia and rotational kinetic energy. This dynamic interplay influences the stability and behavior of the spinning object, making it crucial in understanding various physical phenomena, including gyroscopic motion and the conservation of angular momentum.
  • #1
Trollfaz
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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
 
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  • #2
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.
 
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FAQ: Energy of spinning objects as axis of rotation moves

What is the energy of a spinning object?

The energy of a spinning object is primarily kinetic energy due to its rotation. This rotational kinetic energy is given by the formula \( \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia of the object and \( \omega \) is its angular velocity.

How does the axis of rotation affect the energy of a spinning object?

The axis of rotation significantly affects the moment of inertia \( I \) of the object. If the axis of rotation moves, the distribution of mass relative to the axis changes, which can either increase or decrease the moment of inertia and thus alter the rotational kinetic energy.

What happens to the rotational kinetic energy if the axis of rotation moves closer to the center of mass?

If the axis of rotation moves closer to the center of mass, the moment of inertia \( I \) generally decreases because the mass is more centrally concentrated. As a result, for a given angular velocity \( \omega \), the rotational kinetic energy will decrease.

How is conservation of angular momentum involved when the axis of rotation changes?

Conservation of angular momentum states that if no external torque is acting on the system, the angular momentum \( L = I \omega \) remains constant. When the axis of rotation changes, the moment of inertia \( I \) changes, and consequently, the angular velocity \( \omega \) must adjust to keep the product \( I \omega \) constant, affecting the rotational kinetic energy.

Can the movement of the axis of rotation create or destroy energy?

No, the movement of the axis of rotation cannot create or destroy energy. It can only transform the energy between different forms, such as between rotational kinetic energy and translational kinetic energy, or redistribute it within the system. The total energy of the system remains conserved unless acted upon by external forces.

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