– 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.