Video Transcript
Using the information in the
figure, calculate the coefficient of kinetic friction, rounding the result to the
nearest two decimal places. Given that the mass of the body is
28 kilograms and that the acceleration due to gravity, 𝑔, equals 9.8 meters per
second squared.
Taking a look at our figure, we see
our body which is on some surface, moving along with an acceleration to the right of
2.2 meters per second squared. This acceleration is due to a force
of 155 newtons that’s being applied at an angle of 45 degrees to the horizontal. Knowing this, along with the fact
that the mass of our body — we’ll call it 𝑚 — is 28 kilograms, we want to solve for
the coefficient of kinetic friction between the body and the surface.
We can begin solving for this
coefficient by first sketching in all the forces that are acting on our body. This is called a free-body
diagram. For one thing, we know that our
body is subject to the force of gravity. We also know that there’s an
applied force, we’ll call it 𝐹 sub 𝐴, which is given as 155 newtons, acting at a
45-degree angle on the body. Furthermore, there’s a normal or
reaction force acting straight up from the surface. And lastly, there’s a frictional
force; we’ll call it 𝐹 sub 𝑓. And this opposes the motion of our
object. That’s always true of friction,
which is how we knew that it acted to the left.
Now that we know all the forces
acting on our object, we can recall Newton’s second law of motion. This tells us that the net force
that acts on somebody is equal to that body’s mass times its acceleration. Now because motion in the vertical
direction is independent of motion in the horizontal direction, we can apply
Newton’s second law to either one independently. In other words, we could say that
the net force in the horizontal plane acting on our body is equal to the object’s
mass multiplied by its horizontal acceleration. And indeed, this is the dimension
we’ll focus on because we want to solve for the coefficient of kinetic friction,
which is bound up in the force of friction.
When we say that, we just mean that
the force of friction acting on an object that’s in motion is equal to the
coefficient of kinetic friction multiplied by the normal force acting on the
object. So, here’s what we’ll do. Focusing first on this horizontal
direction, let’s clear a bit of space on screen and apply Newton’s second law in
this horizontal plane. The first thing we’ll do is
establish sign conventions, positive and negative.
Note that the acceleration given to
us in our problem statement is a positive value and that acceleration is to the
right. We’ll say then that any force or
motion in that direction is positive, and any force or motion in the other direction
is negative. As we study the forces acting
horizontally in our free-body diagram, we see that there’s a component of the
applied force that acts horizontally. And then there’s the friction force
which acts entirely in this direction.
Given a 45-degree right triangle
where the hypotenuse is our applied force magnitude, 155 newtons, we can solve for
this horizontal leg of that triangle by multiplying 𝐹 sub 𝐴 by the cos of 45
degrees. That then accounts for the positive
horizontal force acting on our body. We then subtract from that the
frictional force. That force recall is equal to the
coefficient of kinetic friction, what we want to solve for, times the normal
force. This accounts for all of the
horizontal forces acting on our mass. And therefore, by Newton’s second
law, this sum is equal to mass times acceleration in the horizontal direction.
We’re given 𝐹 sub 𝐴 as well as
the mass and acceleration of our body, but we don’t know 𝐹 sub 𝑁 to let us solve
for 𝜇 sub 𝑘. To solve for it, we’re actually
going to need to apply Newton’s second law again, but this time in the vertical
dimension. As we do this, we can set up the
conventions that motion up is in the positive direction and motion down is in what
we’ll call the negative direction. So, let’s consider the vertically
acting forces on our body. First, there’s the normal force, 𝐹
sub 𝑁. Added to that is the vertical
component of our applied force, 𝐹 sub 𝐴.
Returning to our triangle, that
side length is equal to 𝐹 sub 𝐴 times the sin of 45 degrees. And then lastly in the vertical
direction, we have the weight force, 𝑚 times 𝑔, acting downward. By our convention, this force is
negative. And by Newton’s second law, all of
this is equal to the mass of our body times its acceleration in the vertical
direction, what we’ll call 𝑎 sub 𝑣. We then realize that our object
isn’t accelerating vertically, so this value is zero. That means these forces add up to
zero. So, if we subtract this term from
both sides of the equation and add this term to both sides, then we find an
expression for the normal force, which is in terms of the weight force minus the
vertical component of the applied force.
We can then substitute this whole
right-hand side in for 𝐹 sub 𝑁 here. And now, we have an expression we
can use to solve for 𝜇 sub 𝑘, the coefficient of kinetic friction. This is because we know the
numerical values of 𝐹 sub 𝐴, 𝑚, 𝑔, and 𝑎, as well as the sin and cos of 45
degrees. Before we plug in those values
though, let’s rearrange this expression so that 𝜇 sub 𝑘 is the subject. First, we’ll subtract 𝐹 sub 𝐴,
cos of 45 degrees, from both sides. Then, we’ll divide both sides of
the equation by this expression in parentheses. And lastly, we’ll multiply both
sides by negative one. We end up with this expression. And now, we’re ready to substitute
in our values.
𝐹 sub 𝐴 is 155 newtons. Both the cos and sin of 45 degrees
is the square root of two over two. Our mass, 𝑚, is 28 kilograms. The acceleration, 𝑎, is 2.2 meters
per second squared. And lastly, the acceleration due to
gravity, 𝑔, is 9.8 meters per second squared. When we enter this expression on
our calculator, to two decimal places, we find the result of 0.29. This is the coefficient of kinetic
friction between our body and the surface.
