Teaching the Conservation of Angular Momentum

In this article, I appearance how calmly physics problems are apparent back application angular drive conservation. Just starting with an absolute account of angular drive attention allows us to break acutely difficult problems absolutely easily. As always, I use botheration solutions to authenticate my approach.

Again, the bound capabilities of the argument editor force me to use some abnormal notation. That characters is now abbreviated in one spot, the commodity "Teaching Rotational Dynamics".

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Problem. The account (not shown) shows a boy of accumulation m continuing at the bend of a annular belvedere of accumulation M, ambit R, and moment of apathy Ip= (MR**2)/2. The belvedere is chargeless to circle after abrasion about its axial axis. The belvedere is alternating at an angular acceleration We back the boy starts at the bend (e) of the belvedere and walks against its center. (a) What is the angular acceleration of the belvedere back the boy alcove the half-way point (m), a ambit R/2 from the centermost of the platform? What is the angular acceleration back he alcove the centermost (c) of the platform?

Analysis. (a) We accede rotations about the vertical arbor through the centermost of the platform. With the boy a ambit r from the arbor of rotation, the moment of apathy of the deejay additional boy is I = Ip + mr**2. Since there is no net torque on the arrangement about the axial axis, angular drive about this arbor is conserved. First, we account the system's moment of apathy at the three credibility 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 drive at the three points, we have

.................................................Conservation of Angular Momentum

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

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

These aftermost equations are calmly apparent for Wm and Wc in agreement of We:

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

Problem. The account (not shown) shows a compatible rod (Ir = Ml²/12) of accumulation M = 250 g and breadth l = 120 cm. The rod is chargeless to circle in a accumbent even about a anchored vertical arbor through its center. Two baby beads, anniversary of accumulation m = 25 g, are chargeless to move in grooves forth the rod. Initially, the rod is alternating at an angular acceleration Wi = 10 rad/s with the chaplet captivated in abode on adverse abandon of the centermost by latches amid d= 10 cm from the arbor of rotation. back the latches are released, the chaplet accelerate out to the ends of the rod. (a) What is the angular acceleration Wu of the rod back the chaplet ability the ends of the rod? (b) Suppose the chaplet ability the ends of the rod and are not stopped, so they accelerate off the rod. What again is the angular acceleration of the rod?

Analysis. The armament on the arrangement are all vertical and apply no torque about the rotational axis. Consequently, angular drive about the vertical rotational arbor is conserved. (a) Our arrangement is the rod (I = (Ml**2)/12) and the two beads. We accept about the vertical axis

..............................................Conservation of Angular Momentum

......................................(L(rod) + L(beads))i = (L(rod) + L(beads))u

............................((Ml**2)/12 + 2md**2)Wi = ((Ml**2)/12 + 2m(l/2)**2)Wu

so.................................Wu = (Ml**2 + 24md**2)Wi/(Ml**2 + 6ml**2)

With the accustomed ethics for the assorted quantities amid into this aftermost equation, we acquisition that

...........................................................Wu = 6.4 rad/s.

(b) It's still 6.4 rad/s. back the chaplet accelerate off the rods, they backpack their velocity, and accordingly their angular momentum, with them.

Again, we see the advantage of starting every physics botheration band-aid by consulting a axiological principle, in this case the attention of angular momentum. Two acutely difficult problems are calmly apparent with this approach.

Teaching the Conservation of Angular Momentum