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There are non-inertial frames (such as one rotating with the Earth) but the usual laws of kinematics do not have the same form in such frames, and the time dilation derivation you made would not work in rotating coordinates. The Earth's rotation does not cause time to slow down on Alpha Centauri.
I'm trying to understand how rotational energy would change if radius shrinks but mass stays the same. The initial question was posed as follows: "The radius of a disk of matter forming around a new star shrinks by a factor of 2 without any additional mass being added into the system.
As the ball travels (falls) through the air, only if some torque acts on it, will its angular velocity ω ω) change and thus its rotational kinetic energy KR = 1 2Iω2 K R = 1 2 I ω 2 will also change. What happens when the ball hits the ground depends a lot on what the "ground" really is. E.g. let's say the ball hits wet ice (the coefficient ...
Well, considering your body and your arms as one single system on the whole of which gravity is applying torque, we naturally tend to rotate our hands in the same sense as our "rotation of fall" so that maximum of the "change in angular momentum" that gravity gives, which is also in the same sense as the "rotation of fall", goes to the increase in rotation of our arms, and in the process the ...
Since the rod's center of mass is changing, does this mean that it also has translational kinetic energy? Yes. You have both translational kinetic energy and rotational kinetic energy. The translational kinetic energy is due to the translational motion of the COM. Although the rod is rotating about the hinge point, it is also rotating about its ...
Here I am defining the mass of an object (in the usual manner of modern treatments) as the square of that object's energy-momentum four-vector: $$ m \equiv \frac{1}{c^2}|\mathbf{p}|^2 = \frac{1}{c^2}\sqrt{E^2 - (\vec{p}c)^2} \;.$$ Such masses are Lorentz invariants: they do not change under a Lorentz boost (change in inertial velocity).
1. The quick answer is that, for the top to fall over due to gravity, each fragment of the top that is moving around the spin axis has to change its individual direction of movement. They are already changing direction around the spin axis, due the rigidity of the top keeping them moving in a circle.
Let's consider a wheel extended with its axle on both sides. Tie a string at both ends of axle. Note that we should keep the wheel vertical. Upon leaving one string the wheel will tilt. Now keeping the wheel vertical, put the wheel in rotation around the axle and now leave one string. One notices that wheel doesn't fall down as earlier but also ...
For rotational motion, the version of this is the moment of inertia which is similar, but about the tendency to resist angular acceleration. So it is inertia (the moment of inertia if rotation). It keeps rotating at constant angular frequency since it resists a possible change out of nowhere.
A toilet roll with mass m and outer radius R and inner radius r is held by its first piece and is released from a certain height such that it unfurls. The equations of motion, taking downward as positive is given by. mg − T = ma m g − T = m a (Translational) TR = a2R(R2 +r2) T R = a 2 R (R 2 + r 2) (Rotational) If I want upward to be ...