- Good morning!
Bo, what is the equation for arc length?
♫ Flipping Physics ♫
- Arc length equals radius times angular displacement.
- Correct!
Billy, what is the equation for tangential velocity?
- Tangential velocity equals radius times angular velocity.
- Correct!
Bobby, what do you think the equation is
for tangential acceleration?
- Well, it probably equals
the radius times an angular quantity.
I'm guessing it's the radius times the angular acceleration.
- That is correct.
Tangential acceleration equals radius times alpha,
or angular acceleration.
And, for all three of these equations,
you must use radians for the angular quantities.
Let's do an example problem.
Bobby, could you please read the problem
and, Bo, could you please translate?
- A record player is plugged in and uniformly accelerates
to 45 revolutions per minute in 0.85 seconds.
- Angular velocity initial is zero,
because the record player starts out at rest.
The final angular velocity is 45 revolutions per minute,
which I'm sure we need to convert again
to radians per second.
So, multiply by one minute over 60 seconds
and two pi radians over one revolution
to get 1.5 pi radians per second
and change in time is 0.85 seconds.
- Bobby?
- Mints are located 3.0 centimeters, 8.0 centimeters,
and 13.0 centimeters from the center of the record.
- Let's label the three mints, one, two, and three,
and the radii of the paths
of the three mints are 3.0, 8.0, and 13.0 centimeters.
- Bobby, please?
- What is the magnitude of the tangential acceleration
of each mint?
- Tangential acceleration equals question mark
for all three mints.
- Billy, how shall we solve this problem?
- Well, tangential acceleration equals
radius times angular acceleration.
We have the radius of the path for each mint.
However, we do not have the angular accelerations
for all the mints.
- Actually, they are all turning on the record player,
so every mint should have the same angular acceleration.
- And we can solve for the angular acceleration,
using the equation for average angular acceleration.
Average angular acceleration equals
change in angular velocity over change in time
or angular velocity final minus angular velocity initial
over change in time.
So, 1.5 pi minus zero divided by 0.85,
which is 5.54399 radians per second squared.
- We did this before, right?
- Yeah.
- Billy, please, keep going.
- Sure.
Now I can use the equation for tangential acceleration
for the first mint, it is the three times 5.54399,
which is 16.63196 centimeter radians
per second squared.
Um, however, radians have no units and are placeholders,
so they drop out of the units
and we have 17 centimeters per second squared,
with two significant digits.
Um, for the second mint, it is eight times 5.54399,
which is 44.35190
or 44 centimeters per second squared,
again, with two significant digits.
And, for the third mint, it is 13 times 5.54399,
which is 72.07183
or 72 centimeters per second squared.
- Very nice, Billy.
Now, notice the tangential acceleration is, by definition,
tangent to the circle through which the object is moving.
That is what the word, tangential, means in the phrase,
tangential acceleration.
- [Billy] Wow, that's great!
And because the tangential acceleration is tangent
to the circle it is describing,
this means the tangential acceleration is perpendicular
to the radius of the circle.
- [Mr. P] That is correct, Billy.
The tangential acceleration is always perpendicular
to the radius and, with a constant angular acceleration,
the tangential acceleration increases
as the radius increases.
- [Bo] I feel like the two of you just had
that exact same conversation only using the word,
velocity, instead of acceleration.
- [Mr. P] Okay, let's go back to the equation,
arc length equals radius times angular displacement.
Assuming the radius is constant, if we take the derivative
of the arc length equation with respect to time,
we have the derivative of arc length with respect to time
equals r times the derivative of change in angular position
and, actually, we need to take the derivative
of just the angular position with respect to time.
The derivative of arc length with respect to time
is the linear velocity of an object moving along a circle,
which is tangential velocity.
And the derivative of angular position with respect to time
is angular velocity.
So, now we have tangential velocity equals
radius times angular velocity,
which we can also take the derivative of.
Again, assuming that the radius is constant,
we have the derivative of tangential velocity
with respect to time equals radius times the derivative
of angular velocity with respect to time.
The derivative of tangential velocity with respect to time
is tangential acceleration.
And the derivative of angular velocity with respect to time
is angular acceleration.
Therefore, we have tangential acceleration equals
radius times angular acceleration.
- Uh. - Okay.
- I think I understood that. - Uh, wha-ha?
- Now, I know many of you have not taken calculus yet,
so that might have been a little bit
beyond what you understand mathematically.
However, it is still helpful to realize that,
using the derivative,
if you have the equation for arc length,
you can derive the equations for tangential velocity
and tangential acceleration.
For an algebra-based class, you do not need
to understand any of that.
However, it is still useful to get some exposure to it.
Thank you very much for learning with me today.
I enjoyed learning with you.