- Good morning.
If we slowly raise this end of the board
we can solve for the coefficient of static friction
between the book and the incline.
- ♫ Flipping Physics ♫
(book scrapes)
- Bobby, what are our known values?
- Well, it looks like the book starts to slide
when the incline angle is 15 degrees.
I think that's it.
- [Mr. P.] And what exactly are we solving for?
- The coefficient of static friction equals question mark.
- The moment we are interested in
in this demonstration is the moment right before
the book starts to slide.
And the angle of the incline
at that moment is 15 degrees.
(book scrapes)
- Billy, please give us a free body diagram
of all the forces acting on the book
right before the book starts to slide.
- The force of gravity is down,
the board pushes up on the book,
which is the force normal which is perpendicular
to the incline and up.
The force of static friction is parallel
to the incline and opposes the motion of the book.
Without friction the book would slide down the incline
so the force of static friction is up
and parallel to the incline.
- [Mr. P.] And now could you please break
the force of gravity into its components.
- Okay.
Instead of the force of gravity
we can now substitute in its components
the force of gravity parallel,
which is parallel to and down the incline,
and the force of gravity perpendicular
which is perpendicular to the incline and down.
- Bo, could you please do the next step?
- Next we need to sum the forces.
Let's do the parallel direction.
The net force and the parallel direction
equals force of gravity parallel,
which is positive because it is to the right
and force of static friction,
which is negative because it's to the left.
And this also equals mass times acceleration
in the parallel direction,
but I don't know what to do now.
- Bo, that's fine.
Class, what is the acceleration of the book
in the parallel direction before
the book starts to move?
- [All Students] Zero.
- Therefore, before the book starts to move
the force of gravity parallel
and the force of static friction
are equal in magnitude.
So notice as the incline angle increases
the force of gravity down the incline,
what we call the force of gravity parallel, increases.
Therefore, the force of static friction
which adjusts in magnitude to keep the book
from sliding relative to the incline, also increases.
The force of static friction continues
to increase until the magnitude
of the force of gravity parallel
is just barely above the magnitude
of the maximum force of static friction.
At which point the friction switches
from static to kinetic,
decreases in value and the book begins
to accelerate down the incline.
Therefore, we use the maximum force
of static friction in our equation
because the force of static friction
is at its maximum value when the incline
is at 15 degrees.
And because it is static friction
we know the book isn't moving yet,
which means the acceleration in the parallel
direction is equal to zero.
- Bobby, why don't you substitute in some equations, please.
- Force of gravity parallel is
mass times gravity--
- [Bo] Acceleration due to gravity.
- Mass times acceleration due to gravity
times sin theta which equals
force of static friction maximum.
Which is the coefficient of static friction
times force normal.
We need force normal.
- [Bo] Force normal equals force of gravity.
- [Billy] No it doesn't!
- [Bobby] No.
- [Bo] Yes it does.
Force normal equals mass times acceleration
due to gravity.
I've got it in my notes, somewhere.
(paper crinkles)
- Force normal does not equal the force of gravity.
Many students think it does, however it does not.
Now I will admit that often it does;
however, you have to every time
draw a free body diagram and sum the forces
in order to solve for the force normal.
- Bo, please solve for the force normal.
- The net force in the perpendicular direction
equals force normal
minus force of gravity perpendicular.
Which equals mass times acceleration perpendicular.
Acceleration in all directions is zero,
so force normal equals force of gravity perpendicular.
Which equals mass times acceleration due to gravity
times cosine theta.
Pretty much equals force of gravity
it's just force of gravity perpendicular instead.
It doesn't really matter.
- Ooo.
Billy, please continue.
- We can substitute for force normal
mass times acceleration due to gravity
times cosine theta, into the equation
we had from the net force and the parallel direction.
We now have mass times acceleration due to gravity
times sine theta equals the coefficient
of static friction times mass times acceleration
due to gravity, times cosine theta.
Everybody brought mass to the party!
- [Bobby] Actually, everybody brought
the force of gravity to the party.
(upbeat, funky digital music)
♫ Everybody brought mass ♫
- And we can divide both sides by the cosine
of the incline angle and the sine of theta over cosine theta
equals tangent theta.
Therefore, the coefficient of static friction
between the book and the incline
is equal to the tangent of the incline angle
right before the book starts to slide.
Pretty cool, hey?
- [Billy] That is pretty cool.
- [Bobby] Yeah. - [Bo] I guess.
- We can plug in 15 degrees for our incline angle
and we get that the coefficient of static friction
between the book and the incline
is 0.267949 or 0.27
with two significant digits.
Thank you very much for learning with me today.
(chuckles) I enjoyed learning with you.