9.1 Uniform Circular Motion

MIT 8.01 Classical Mechanics, Fall 2016
Instructor: Dr. Peter Dourmashkin

License: Creative Commons BY-NC-SA

[JoséAntonioBottino]

In a Uniform Circular Motion, the linear speed (tangential speed) v remains constant.

If the object makes n revolutions (cycles) in a time t, then it travels a distance s
s = 2 • 𝜋 • r • n

where n is the “number of revolutions”, n is dimensionless, n has unit rev/rev = 1.

Since v = s / t, then
v = (2 • 𝜋 • r • n) / t

Since v = ω • r, then
ω • r = (2 • 𝜋 • r • n) / t.

This implies that
ω = (2 • 𝜋 • n) / t

If ω = 2 • 𝜋 • f, where f is the frequency, then.
2 • 𝜋 • f = (2 • 𝜋 • n) / t.

This implies that
f = n / t

or what is the same, the frequency f is the number of revolutions (cycles) per unit time (usually seconds).

The unit of f should be
(rev/rev)/s = Hz = 1/s

equal to the number of revolutions per second [nrps = (rev/rev)/s, if the custom is to be maintained], and not in revolutions per second (rps = rev/s).

The unit hertz (Hz) replaced the unit cycles per second, which was actually the number of cycles per second.

Given that the period T = 1 / f, then
T = t / n.

Since the period T is the time it takes for the object to complete one revolution (one cycle), then the unit of T is:
s/(rev/rev) = s

equal to seconds per number of revolutions (second per number of cycles).

As
ω = θ / t

and θ is the number of radians, θ is dimensionless, θ is measured in rad/rad = 1, that means that ω must be measured in
(rad/rad)/s = 1/s = s^(-1)

and not in rad/s.

It is understood that in the formula
ω = 2 • 𝜋 • f

the unit conversion is
1 (rad/rad)/s = 2 • 𝜋 • (rev/rev)/s

so
1 (rad/rad) = 2 • 𝜋 • (rev/rev).

There the 2𝜋 allows us to go from “number of revolutions” (rev/rev) to “number of radians” (rad/rad).

I will highlight the difference between the unit of angular speed, which seems to be 1/s and the unit of frequency
which also appears to be 1/s. They are different. Hertz is number of revolutions per second (nrps) while angular speed
is the number of radians per second (nrad/s, stretching the notation a bit).

[changtillend]

Thank you so much... This is Very Very Helpful Who need this...

[billaliqbal]

How does he manage to write backwards? I've been unable to sleep without knowing. Is this the MIT big brain?

[abhishektiwari]

never seen that invisible kinda black board ever.
well, it is super cool.