- Good morning.
We are going to determine the coefficient
of kinetic friction between the book and the incline,
given the following information.
♫ Flipping Physics ♫
- [mr.p] Bo, please tell me what we know.
- Well, it looks like the angle
of the incline theta is 14 degrees,
the change in time is 2.05 seconds,
and the displacement of the book is 0.78 meters.
Oh, and the initial velocity of the book is zero,
and the coefficient of kinetic friction between
the book and the incline equals question mark.
- Good.
Because the displacement is neither in the x direction
nor the y direction, but rather parallel to the incline,
I have labeled the displacement delta D parallel
for the displacement parallel to the incline.
In addition to that, because the book is moving parallel
to the incline, I have labeled the initial velocity,
the initial velocity parellel which is equal to zero.
Bobby, what should we do next?
- Draw the free body diagram.
- [mr.p] Yes, please.
Draw the free body diagram.
- Okay, the force of gravity is straight down,
the force normal is up and perpendicular to the incline,
and the force of kinetic friction is parallel
to the incline and opposes the sliding of the book,
so up the incline.
- And we can break the force of gravity
into its components and redraw the free-body diagram.
So now instead of the force of gravity straight down,
we have its components.
The force of gravity perpendicular,
which is perpendicular to the incline and down,
and the force of gravity parallel,
which is parallel to the incline and down.
So, we have drawn our free body diagram,
we've broken forces into components,
and we've redrawn the free body diagram.
Billy, what should we do next?
- Let's sum the forces.
The net force in the perpendicular direction
equals force normal minus force of gravity perpendicular,
which then equals mass times the acceleration
in the perpendicular direction.
The book is not moving perpendicular to the incline,
so the acceleration in the perpendicular direction is zero,
therefore the force normal equals the force
of gravity perpendicular and the equation
for the force of gravity perpendicular
is mass times acceleration of gravity times the cosine
of the incline angle, and we can put that
in our equation holster.
- Bobby, please keep going.
- Now we can sum the forces in the parallel direction,
the net force in the parallel direction equals force
of kinetic friction minus force of gravity parallel
which equals mass times acceleration
in the parallel direction.
- We can substitute the coefficient
of kinetic friction times force normal
for the force of kinetic friction.
- And mass times acceleration due
to gravity times sine theta for the force
of gravity parallel.
- And then from our equation holster,
we can substitute mass times acceleration
due to gravity times cosine theta for the force normal,
we now have the coefficient of kinetic friction
times mass times acceleration
due to gravity times cosine theta
minus mass times acceleration due to gravity
times sine theta equals mass times acceleration parallel.
- Everybody brought mass to the party.
(funky electronic music)
♫ everybody brought mass ♫
- We can add the acceleration due to gravity
times the sine of theta to both sides
and then divide both sides by the acceleration
due to gravity times the cosine of theta
to get the coefficient of kinetic friction
equals the acceleration due to gravity
times the sine of the incline angle
plus the acceleration in the parallel direction
all divided by the acceleration due to gravity
times the cosine of the incline angle.
And, Bo, what is the one variable we still need
in this equation in order to solve
for the coefficient of kinetic friction?
- The acceleration in the parallel direction.
- [mr.p] And how are we going to find the acceleration
in the parallel direction?
We can use a uniformly accelerated motion equation.
- [mr.p] That is correct.
Please do so.
- We can use the equation displacement parallel
equals velocity initial parallel times change
in time plus one half times acceleration parallel
times change in time squared.
The initial velocity is zero so that term cancels out.
We can multiply by two and then divide by change
in time squared to get acceleration parallel
equals two times displacement parallel
divided by change in time squared.
With numbers that is two times 0.78
divided by 2.05 squared which is 0.371208 meters
per second squared, and we can substitute that number
into the equation we have for the coefficient
of kinetic friction which equals 9.81 times sine
of 14 plus 0.371208 divided by 9.81 times cosine
of 14 which is 0.288326 or 0.29 with two significant digits.
Wait a second, last time didn't we get 0.27
for the coefficient of static friction between that book
and that incline?
- Dang and blast, that doesn't make any sense.
The coefficient of kinetic friction
is supposed to be less than the coefficient
of static friction.
- That is correct.
There is a mistake in this solution.
- The acceleration should be negative
because the book is speeding up in a negative direction.
- Right.
But that comes from the fact that the displacement
in the parallel direction should be negative 0.78 meters.
As you said, the book is moving in a negative direction.
- Yes, the displacement is negative.
Notice when we sum the forces in the parallel direction,
the force of gravity parallel was negative
because we said down and to the left was negative.
And that makes the acceleration negative,
which makes the coefficient of kinetic friction 0.21
which is, as it should be, less than the coefficient
of static friction we got before of 0.27.
You have to be very careful to remember
which directions are positive
and which directions are negative, please.
Thank you very much for learning with me today.
I enjoyed learning with you.