Understanding The Concept Of Angular Velocity

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In this article, I display how without difficulty physics troubles are solved while the use of Angular Velocity Calculator. Just beginning with an express declaration of angular momentum conservation permits us to remedy apparently hard troubles pretty without difficulty. As always, I use trouble answers to illustrate my approach.

Again, the constrained skills of the textual content editor pressure me to apply a few uncommon notation. That notation is now summarized in a single spot, the article "Teaching Rotational Dynamics".

Problem. The sketch (now no longer shown) indicates a boy of mass m status at the brink of a cylindrical platform of mass M, radius R, and second of inertia Ip= (MR**2)/2. The platform is loose to rotate with out friction round its important axis. The platform is rotating at an angular speed We while the boy begins offevolved at the brink (e) of the platform and walks closer to its middle. (a) What is the angular speed of the platform while the boy reaches the half-manner point (m), a distance R/2 from the middle of ...
... the platform? What is the Angular Velocity Calculator while he reaches the middle (c) of the platform?

Analysis. (a) We bear in mind rotations across the vertical axis via the middle of the platform. With the boy a distance r from the axis of rotation, the instant of inertia of the disk plus boy is I = Ip + mr**2. Since there may be no internet torque at the machine across the important axis, angular momentum round this axis is conserved. First, we calculate the machine's second of inertia on the 3 factors of interest:

...................................... EDGE.............Ie = (MR**2)/2 + mR**2 = ((M + 2m)R**2)/2

...................................... MIDDLE..........Im = (MR**2)/2 + m(R/2)**2 = ((M + m/2)R**2)/2

.......................................CENTER..........Ic = (MR**2)/2 + m(0)**2 = (MR**2)/2

Equating the angular momentum on the 3 factors, we have

.................................................Conservation of Angular Velocity

..........................................................IeWe = ImWm = IcWc

...................................((M + 2m)R**2)We/2 = ((M + m/2)R**2)Wm/2 = (MR**2)Wc/2

These final equations are without difficulty solved for Wm and Wc in phrases of We:

.....................................Wm = ((M + 2m)/(M + m/2))We and Wc = ((M + 2m)/M)We.

Problem. The sketch (now no longer shown) indicates a uniform rod (Ir = Ml²/12) of mass M = 250 g and duration l = a hundred and twenty cm. The rod is loose to rotate in a horizontal aircraft round a set vertical axis via its middle. Two small beads, every of mass m = 25 g, are loose to transport in grooves alongside the rod. Initially, the rod is rotating at an angular speed Wi = 10 rad/s with the beads held in region on contrary facets of the middle with the aid of using latches positioned d= 10 cm from the axis of rotation. When the latches are released, the beads slide out to the ends of the rod. (a) What is the angular speed Wu of the rod while the beads attain the ends of the rod? (b) Suppose the beads attain the ends of the rod and aren't stopped, in order that they slide off the rod. What then is the Angular Velocity of the rod?