Video Transcript
A body of mass 74 kilograms was
projected at 8.5 meters per second along the line of greatest slope up a plane
inclined at 30 degrees to the horizontal. Given that the resistance of the
plane to its motion was 7.4 newtons, find the distance the body traveled until it
came to rest. Take 𝑔 to equal 9.8 meters per
second squared.
Alright, so let’s say that this is
our slope with its inclination angle of 30 degrees. And on this slope, we have a body
being projected along the slope line at 8.5 meters per second. As the body moves up the slope, it
encounters a frictional force we can call 𝐹 with a given magnitude of 7.4
newtons. We’re told further that the body
has a mass we’ll call 𝑚 of 74 kilograms. And knowing that the body
eventually comes to a rest, we want to calculate the distance along the slope, we’ll
call it 𝑑, that it takes to do this.
As we get started, let’s clear a
bit of space on screen and recall that Newton’s second law of motion tells us that
the net force acting on a body is equal to its mass times its acceleration. Considering an up-close view of our
body as it moves up the incline, let’s draw a free-body diagram showing all the
forces acting on it. We know that the body experiences a
weight force, its mass times the acceleration due to gravity, a normal or reaction
force perpendicular to the surface of the plane. And because of the body is moving
uphill, it experiences a frictional force downhill. These are the forces that act on
our body as it rises up the incline.
And let’s say that, directionally,
the positive 𝑥-direction points down the incline and the positive 𝑦-direction
points perpendicularly away from it. Applying Newton’s second law to
this scenario, we can say that the sum of forces in the 𝑥-direction equals our
body’s mass times its acceleration in this direction. These forces include the frictional
force 𝐹, as well as a component of the weight force that acts in the
𝑥-direction.
To solve for this component, we
need to realize that this angle in our right triangle here is identical to the
30-degree angle of inclination of our plane. Because that’s true, the component
of the weight force we’re interested in is 𝑚 times 𝑔 times the sin of 30
degrees. The sin of 30 degrees is
one-half. So, when we write out all the
forces acting on our body in the 𝑥-direction, we have 𝐹 plus 𝑚 times 𝑔 divided
by two. The second law tells us that this
sum equals our body’s mass times its acceleration in-in this case, the
𝑥-direction. Dividing both sides of this
equation by 𝑚, we find that factor cancelling on the right. And we find that 𝑎 sub 𝑥 is equal
to 𝐹 plus 𝑚 times 𝑔 over two all divided by the mass 𝑚.
Since we’re given the force 𝐹, the
mass 𝑚, and we also know that the constant 𝑔 is equal to 9.8 meters per second
squared, we can substitute those values into this equation to solve for 𝑎 sub
𝑥. When we enter this expression on
our calculator, we find a result of exactly five with units of meters per second
squared. This means that as our body moves
up the incline, it decelerates at 5 meters per second squared. And this will help us solve for the
distance 𝑑 it takes to come to a rest.
Because our body experiences a
constant acceleration, that means its motion can be described by what are called the
equations of motion. These are sometimes also called the
kinematic equations or SUVAT equations. And in our case, we’ll look at a
specific equation of motion that tells us that the final velocity of a body squared
equals its original velocity squared plus two times its acceleration multiplied by
its displacement. Rearranging this equation to solve
for 𝑑, it’s equal to 𝑣 sub 𝐹 squared minus 𝑣 sub zero squared all over two
𝑎.
And as we consider our scenario
with respect to these variables, we know that 𝑣 sub 𝐹 is zero because our body
ends up at rest. We’re told the value of what’s
called here 𝑣 sub zero. That’s 8.5 meters per second. And we’ve just recently solved for
the acceleration 𝑎. If we plug in these values and
leave out units for now, we get this expression. And notice that it’s a negative
value. The reason for this is that,
technically, we’re solving for displacement, which, according to our sign
convention, where positive values are down the incline is actually truly a negative
value.
What we actually want to solve for,
though, is the distance that our body travels before it comes to rest. To calculate that in this case, we
just need to take the absolute value of this fraction. Entering this expression on our
calculator, we get a result of 7.225. The units for this distance are
meters. So, we can say that this body moves
7.225 meters up this incline before it comes to a stop.
