- [Voiceover] Good morning.
This is Billy again, and this is another video project for
♫ Flipping Physics ♫
Mr. P asked me to show that the
Conservation of energy problem
with friction, an incline and a spring, which I did,
can also be solved using the work due to friction
equals change in mechanical energy equation,
so here goes.
We have the same problem.
A block with a mass of 11 grams
is used to compress a spring a distance of 3.2 centimeters.
The spring constant of the spring is 14 Newtons per meter.
After the block is released,
it slides along a level, frictionless surface
until it comes to the bottom of a 25 degree incline.
If the coefficient of kinetic friction
between the block and the incline is 0.30,
to what maximum height does the block slide?
Again, we list our known values and convert to base SI units
but now we need to use the work due to friction
equals change in mechanical energy equation.
We can use this equation because there is work done
by a force of friction.
However, there is no work done by a force applied.
Expanding out the equation gives us the work due to friction
equals the force of kinetic friction
times the displacement of the block
times the cosine of the angle between those two vectors.
And the change in mechanical energy
equals mechanical energy final
minus mechanical energy initial.
Notice the force of friction only acts on the block
while it is on the incline,
so the work due to the force of friction
is only while the block is on the incline.
The angle between the force of kinetic friction,
which is down the incline,
and the displacement of the block,
which is up the incline, is 180 degrees.
Let's set our initial point to where the spring
is at its maximum compression, and let's set the final point
to where the block is at its maximum height.
Also, let's set the horizontal zero line
at the center of mass of the block
while it is on the horizontal surface.
Now let's identify the mechanical energies.
The initial velocity of the block is zero,
so there is no initial kinetic energy.
The initial height of the block is zero,
so there is no initial gravitational potential energy.
There is an initial compression of the spring,
so their is elastic potential energy initial.
The final height of the block is not zero,
so there is gravitational potential energy final.
The final velocity of the block is zero,
so there is no kinetic energy final.
The block is not compressing the spring at the end,
so there is no elastic potential energy final.
Substituting in the equation
for the force of kinetic friction
gives us the coefficient of kinetic friction
times force normal times the displacement
in the parallel direction times the cosine of 180 degrees,
which equals mass times acceleration due to gravity
times height final minus one-half times the spring constant
times the initial compression of the spring squared.
Let's put that in our equation holster
because we need to find the force normal.
Let's draw a free body diagram
and sum the forces in the perpendicular direction.
The force normal is up and perpendicular to the incline.
The force of gravity is straight down,
and the force of kinetic friction is parallel to the incline
and opposite the direction of motion or down the incline.
We can break the force of gravity into its components:
the force of gravity perpendicular to the incline and down
and the force of gravity parallel to and down the incline.
The net force in the perpendicular direction
equals force normal minus force of gravity perpendicular,
and the net force in the perpendicular direction
also equals mass times the acceleration
in the perpendicular direction.
The block isn't moving perpendicular to the incline,
so the acceleration perpendicular is zero,
and therefore the force normal
equals the force of gravity perpendicular,
which equals mass times acceleration due to gravity
times the cosine of the incline angle,
which we can substitute back into the equation
in our equation holster.
Oh, and we can also substitute in negative one
for the cosine of 180 degrees.
Other than the height final,
which is the maximum height we are solving for,
the only variable we do not have in this equation
is the displacement in the parallel direction.
So we can draw a triangle
and get the sine of the incline angle
equals opposite over hypotenuse.
Opposite the incline angle is the height final,
and the hypotenuse is the displacement
in the parallel direction.
Rearranging the equation gives us
the displacement in the parallel direction
equals the height final
divided by the sine of the incline angle,
which we can substitute back into the equation
we were just working with.
Because sine theta divided by cosine theta
equals tangent theta,
we now have negative coefficient of kinetic friction
times mass times acceleration due to gravity
divided by tangent theta all times height final
equals mass times acceleration due to gravity
times height final minus one-half times the spring constant
times the initial compression of the spring squared.
Then we multiply through by negative one
and move gravitational potential energy
over to the other side so we can take out the common factor
of mass times acceleration due to gravity
times height final.
Then we can solve for the height final
by dividing both sides
by mass times acceleration due to gravity times
the quantity one plus the coefficient of kinetic friction
divided by the tangent of the incline angle.
Now we can substitute in numbers.
The height final equals one-half times 14
times 0.032 squared
divided by the quantity 0.011
times 9.81 times the quantity one plus 0.3
divided by the tangent of 25 degrees,
which works out to be, with two significant digits,
0.040 meters,
which is exactly what we got the previous time
we did this problem.
The maximum height the block slides to
is 0.040 meters.
Thank you very much for learning with me today.
I enjoyed learning with you.