Question

Physics/Chemistry questions: Rotation questions:
Can anyone explain to me
a) how is the formula for centripetal acceleration (a = v^2/r) deduced or derived?
b) What is the difference between the formula for angular momentum (L = rmv) versus saying L = Iw Are these both the same and how are they related?
c) how is the chemistry formula for the angular momentum of an electron (L=nh/2pi) derived from the general formula for angular momentum?

Answer 1

(a) can be derived by considering the position of a particle moving in a circle of radius R about an origin with speed v:
\[ <(x(t),y(t)> = <R \cos(\omega t),R\sin(\omega t)>\]
The acceleration is then the second derivative:
\[ <a_x(t),a_y(t)> = -\omega^2R <\cos(\omega t),\sin(\omega t)> \]
The magnitude of this acceleration is
\[ |a| = \omega^2 R\]
and since
\[\omega = \frac{2\pi v}{2\pi R} = \frac{v}{R} \]
we have
\[ |a| = \omega^2 R = \frac{v^2}{R} \]

Answer 2

(b) if you're talking about a point particle moving around in a circle, then
\[ L = I\omega = (mR^2)(\frac{v}{R}) = mvR \]
However, the moment of inertia
\[I\]
can be applied to composite systems, i.e. disks and spheres and other objects that cannot be considered points. We calculate it as follows:
\[I = \int \rho r^2 dV \]
In that sense, the moment of inertia of a composite object about some point (mR^2) is the sum of the angular momenta of its constituent particles about that point.

Answer 3

Finally, (c) cannot really be derived from anywhere unless you're prepared to go into a semester's worth of Quantum Mechanics. The reduced Planck's constant, h/ 2 pi, is in units of Joule * seconds, or
\[ kg \frac{m^2}{s^2} \cdot s = kg\frac{m^2}{s} = kg\frac{m}{s} \cdot m\]
Which is the same unit as angular momentum -> we can consider it a natural unit of angular momentum, and so Bohr postulated that the angular momentum of the electron about the nucleus must be some integer multiple of h/ 2pi.

Answer 4

a): A nice (but hardly rigorous) 'proof' is that for a larger speed the force must pull in that ratio more, but as it moves faster, it must pull in more times per second, hence the v^2. r is obvious as if it is further away, then it needs to pull in less per second, as the trajectory is closer to a straight line.

Answer 5

Awesome help! Jemurray3 that resolves my confusion about the derivations. Henpen, that is a very nice way to visualize it. Thanks!