– 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.