- [Voiceover] Good morning!
This is Billy and again, this is another video project for
♫ Flipping Physics ♫
Mr. P asked me again to show
that the conservation of energy problem
with friction an incline and a spring,
which I did twice,
can also be solved using the work-energy theorem
or what Mr. P prefers to call
the net work-kinetic energy theorem
because evidently we tend to confuse
the net work equals change in kinetic energy equation
with the work due to friction equals change
in mechanical energy equation.
Again, the example is,
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?
We still list our known values
and convert to base SI units,
but now we are going to use the net work
kinetic energy theorem, so net work equals the change
in kinetic energy. Set the initial point
where the spring is compressed to its maximum distance,
and the final point at the maximum height
to which the block moved.
The block is not moving at the initial point,
so the initial kinetic energy is zero.
And the block is also not moving at the final point,
so the final kinetic energy is also zero.
Therefore, the net work equals zero minus zero
which is zero.
The equation for work is work equals the magnitude
of the force doing the work on the objects
times the magnitude of the displacement
of the objects times the cosine of the angle
between those two vectors.
Because the direction of the displacement of the block
changes, we need to split the net work into two parts,
the net work on the horizontal frictionless surface
and the network on the incline.
Let's start with the net work
on the horizontal frictionless surface.
In order to do so, we need a free body diagram
of the forces acting on the block
while it is being pushed by the spring
on the horizontal surface.
Force normal is straight up.
The force of gravity is straight down,
and the force of the spring is to the right.
The direction of the force normal
is straight up, and the direction
of the force of gravity is straight down.
Therefore, the angle between the direction
of the displacement of the block, which is to the right,
and those two forces is the same.
It is 90 degrees.
Therefore, because the cosine of 90 degrees is zero,
the work done by both the force normal
and the force of gravity on the block
while it is on the horizontal surface
are both zero.
Because the force caused by the spring
is not constant, we need to solve
for the work done by the force of the spring
a bit differently.
As the block is being pushed by the spring,
the spring loses elastic potential energy.
That loss in elastic potential energy
equals the work done by the force of the spring
on the block.
In other words, the work done by the spring
equals the negative of the change in elastic potential
energy of the spring, or the negative of the quantity
of the final elastic potential energy
minus the initial elastic potential energy.
Because the block is not on the spring
at the final position,
the elastic potential energy final is zero.
Therefore, the work done by the spring on the block
equals the elastic potential energy initial of the spring.
The equation for which is one half times the spring constant
times the initial compression of the spring squared.
The net work done on the block
while it is on the horizontal surface
is then the sum of the work done
by the force normal,
the work done by the force of gravity,
and the work done by the force of the spring.
Therefore, because two of those are zero,
the net work done on the horizontal
is just the work done by the force of the spring.
Now let's move onto the work done
on the block while it is on the incline.
Again, we need to draw a free body diagram.
The force normal is perpendicular to the incline and up.
The force of gravity is straight down,
and the force of kinetic friction is parallel
to the incline and opposite of the direction
of the motion of a block, so the force of kinetic friction
is down the incline.
Again, the work done by the force normal
is zero because the direction between the force normal
and the displacement of the block
is 90 degrees.
The work done by the force of gravity
equals the force of gravity times the displacement
of the block in the parallel direction
times the cosine of the angle
between those two directions.
There are going to be several angles
in this problem, so I have labeled this angle theta one.
Theta one is the angle between the displacement of the block
which is up the incline, and the force of gravity
which is straight down.
Theta one is 90 plus 25 or 115 degrees.
In a previous video, I did a detailed explanation
of that angle, so if you need a better understanding
of why theta one is 115 degrees,
click on the link that just popped up
or the one in the description of this video.
We now have an equation for the work done
by the force of gravity.
It is equal to mass times the acceleration due to gravity
times the displacement of the block
in the parallel direction times the cosine
of theta one,
and theta one is 115 degrees.
Oh, but we need the displacement in the parallel direction
in terms of what we are solving for, height maximum.
We can draw a triangle to get the sine
of the incline angle, which we will label theta two
equals opposite over hypotenuse, which is height maximum
divided by displacement parallel.
We can rearrange that equation to get displacement
parallel equals height maximum divided by
the sine of theta two.
Substituting back into the work
due to the force of gravity equation
gives us the work due to the force of gravity
equals mass times acceleration due to gravity
times height maximum times the cosine
of theta one, all divided by the sine of theta two
where theta two is 25 degrees, the incline angle.
The only force doing work on the block
which we have yet to solve for
is the force of kinetic friction.
So, the work done by the force of kinetic friction
equals the force of kinetic friction times displacement
in the parallel direction times the cosine of another angle
which I have labeled theta three.
We can substitute the coefficient of kinetic friction
times force normal for the force of kinetic friction
and height maximum divided the sine of theta two
for the displacement in the parallel direction.
Theta three is the angle between the direction
of the displacement of the block
which is up the incline
and the direction of the force of kinetic friction
which is down the incline.
Therefore, theta three is 180 degrees.
We need force normal.
So, we need to 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.
After re-drawing the free body diagram,
we can sum the forces in the perpendicular direction.
The net force in the perpendicular direction
equals the force normal minus
the force of gravity perpendicular.
The net force in the perpendicular direction
also equals mass times the acceleration
in the perpendicular direction.
The block is not moving perpendicular to the incline,
so the acceleration in the perpendicular direction
equals zero.
Therefore, the force normal equals the force
of gravity perpendicular, which equals mass
times the acceleration due to gravity
times the cosine of the incline angle
which is theta two.
We can substitute that back into the equation
for the work done by the force of friction
to give us work due to the force of kinetic friction
equals the coefficient of kinetic friction
times mass times acceleration due to gravity
times the cosine of theta two
times the quantity height maximum
divided by the sine of theta two,
all times the cosine of theta three
because cosine theta divided by sine theta
equals one over tangent theta.
We can substitute the tangent of theta two
into the equation.
Now we can go back to the original
net work kinetic energy theorem.
Net work equals the change in kinetic energy,
which in this problem is zero,
which equals the sum of the work done
by the force of the spring,
the work done by the force of gravity,
and the work done by the force of friction.
We have equations for all of those
which we can substitute in.
Zero equals one half times the spring constant
times the initial compression of the spring squared
plus the quantity mass times acceleration due to gravity
times height maximum times cosine of theta one,
all divided by the sine of theta two
plus the quantity the coefficient of kinetic friction
times mass times acceleration due to gravity
times height maximum times the cosine of theta three
all divided by the tangent of theta two.
Substituting in numbers gives us one half
times 14 times 0.032 squared
plus the quantity 0.011
times 9.81 times height maximum
times cosine of 115 all divided by sine of 25
plus the quantity 0.30 times
0.011 times 9.81
times height maximum times cosine of 180,
all divided by the tangent of 25.
Let's calculate some numbers.
Zero equals 0.007168
plus negative 0.10791
times height maximum plus
negative 0.069424 times height maximum.
Adding the two height maximum terms together
and moving them over to the other side
of the equation gives us
0.177334 times height maximum equals
0.007168 and dividing by
0.177334 gives us
0.040421 or 0.040 meters
with two significant digits,
which is the exact same answer we got
the other two times we did this problem.
You know what? Let's not actually be done there
because I think we can learn more from these numbers,
which is probably why Mr. P asked me
to do this problem in the first place.
Notice the work done by the force of the spring,
which is approximately 0.072 joules
is positive
The work done by the force of the spring is positive
because the spring converts all the initial
elastic potential energy into kinetic energy.
The work done by the force of kinetic friction
is negative 0.069424
times the height maximum, or approximately
negative 0.0028 joules.
The work done by the force of kinetic friction
is negative because it converts kinetic energy
to heat and sound energy.
The work done by the force of gravity
is negative 0.10791
times the height maximum,
or approximately negative 0.0044 joules.
The work done by the force of gravity
is negative because it converts kinetic energy
into gravitational potential energy.
In fact, if we look at the equation for the work done
by the force of gravity, it equals mass
times acceleration due to gravity times height maximum
times the cosine of theta one all divided by
sine of theta two.
Substituting in values for theta one and theta two
gives us the cosine of 115 divided by the sine of 25
which is negative one.
This shows the work done by the force of gravity
on the block equals the negative
of the gravitational potential energy gained by the block
which is pretty cool.
Let's actually go all the way back to the beginning
to understand this better.
Before we apply a force to the block
to compress the spring,
the velocity of the block is zero, so the kinetic energy
of the system is zero.
The spring is at its equilibrium position,
so the elastic potential energy is zero
and the block is at the horizontal zero line,
so there is no gravitational potential energy.
In other words, before we apply a force to the block
to compress the spring, the system has zero
mechanical energy.
Then we apply a force to the block
to compress the spring and the work done
by the force applied adds energy to the system
in the form of elastic potential energy.
When we let go of the block,
the block slides along the frictionless surface.
The force of the spring does work on the block
to convert all the elastic potential energy
to kinetic energy.
Then the block slides up the incline, which has friction.
Some of the kinetic energy is converted to
gravitational potential energy by the force of gravity
and the remaining kinetic energy is converted to heat
and sound by the force of friction.
We can see that in terms of numbers
because if we take the work done
by the force of kinetic friction divide it by the initial
energy of the system and multiply by 100,
we get approximately negative 39 percent,
which is the percentage of the initial
elastic potential energy of the system
which was dissipated as heat and sound energy,
and if we take the work done by the force of gravity
divide it by the initial energy of the system
and multiply by 100, we get approximately
negative 61 percent, which is the initial
elastic potential energy of the system
which was converted to gravitational potential energy.
And that adds up to 100 percent
which it absolutely should.
Pretty cool stuff!
I'm sorry this took so long.
I think it was totally worth it.
I hope you think so, too.
Thank you very much for learning with me today.
I enjoyed learning with you.