in this video we have a block sliding down a smooth ramp in other words we don't have
to think about friction we're given the angle of incline there and the first thing we want to do
is find the speed of the block just before it hits a spring that's laying on the ramp
and we're going to use energy conservation for that and then in the second question we want
to get the maximum compression of that spring and that means the compression of the spring right at
the moment where it brings the block to a stop and we're going to use energy for that question too
it's just some of the energy's spring energy so the complication in this question is really that
there's some trig to do along with the energy conservation concepts and when i look at part
a i want the speed of the block right before it hits the spring well that depends on the initial
height of the block but what we have is the distance between the spring and the block that's
a hypotenuse of a right triangle and what we want is the vertical leg to get that initial height
so we'll go ahead and draw the right triangle real quick and that also has an angle of 34 degrees
the hypotenuse is one and a half meters and that means the length of this vertical side is 1.5
sine of 34. i'll go ahead and get a decimal approximation on that and it comes out to 0.839
meters so then i set up my energy conservation equation and in this case it's just gravitational
potential energy turning into kinetic energy setting the zero of my height right here
at my final height and that keeps things as simple as possible so we write e initial equal d final
we don't have to worry about any non-conservative forces acting here and my initial energy well the
block is stationary and it has some height above the zero point so it's all potential energy in my
final energy well the spring is not compressing yet and the height is zero so it's all kinetic so
i'll call that one half mv final squared the m's are going to cancel out and when i solve
for v finals symbolically i get square root 2g y initial then we just have to plug in the numbers
and i get 4.05 meters per second part b is the tricky part of the problem partly because the
setup is more difficult and partly because the math is more difficult so i'll go ahead and draw
the block in the final state that we're referring to in part b i want the maximum compression of the
spring and this means the spring has brought the mass to a stop the spring is now more compressed
than it used to be since i don't know what the compression length is that's what i'm trying to
solve for i'm going to go ahead and call that d now in this final state there's no kinetic energy
because this is the turning point where the spring reaches maximum compression this is just before it
starts to expand and shoot the block back up the ramp so i don't have to worry about kinetic energy
but the potential energy is a little tricky if i use the same zero for potential energy
then my final height for this block is going to be listed as a negative number
notice that i still have a triangle similar to the original triangle that angle is 34 degrees
and the length of this vertical side is d sine theta so that's a magnitude for that
but the actual final y coordinate is negative d sine theta where d is presumed to be a positive
number so that negative d sine theta is a negative number so we set up the energy conservation
equation e initial equals z final and to be clear i'm referring to a different final state
this time now i'm talking about the moment of max compression instead of the moment just before it
hits the spring and in my initial state there's no motion there's no spring compression it's just the
gravitational potential energy corresponding to that initial y coordinate so we'll call it
mgy initial and in my final state i have both a gravitational potential energy term mgy final
where again y final is negative and i have some energy in the compression of
the spring so that's one half k d squared because we used a d for the compression
so i can go ahead and throw some numbers into this now the mass was 2 kilograms
g is 9.8 initial y value to three sig figs was 0.839 meters above the zero point that we
chose my final gravitational potential energy term has a final y-coordinate that must be a
negative number because it's below the zero so we took care of that here using a positive d
with a minus sign out in front of it and then finally we have our spring energy term that's one
half times k which is 50 in this problem times d squared so this is a non-trivial quadratic
equation i have a quadratic term and a linear term and a constant term and i've got to solve for d
and to set this thing up i want to clean up the numbers a little bit so like one half of 50 is 25
and i get 25 d squared to start things out so what i'm doing here is reversing
the direction of this equation so that it looks more like the setup for the quadratic formula
i have 25d squared and then my next term is negative and 2 times 9.8 times the sine of 34
turns out to be 10.96 so i end up with a negative 10.96 times d and then this other
term it has to be moved over to the other side of the equation before i flip the equation around
it gets a minus sign as a result and 2 times 9.8 times 0.839 turns out to be 16.44
and now i'm set up in the right form to apply the quadratic formula so especially in my life
sciences physics classes i don't see any problem with using a computer utility for this because so
many people already have a quadratic formula algorithm in their calculator so in my case
i plugged into a computer algebra system and i get two values of d and it's either 1.06 meters
or negative 0.62 meters and this is why it's so key to be clear about the fact
that we were using a positive d right from the beginning and i put that into the mathematics
by making sure to indicate i had a negative final y coordinate by putting a minus sign in front of
a positive d and i see it here again where i know my final gravitational potential energy
must be a negative number because i put the zero of heights right here and for that to be negative
d must be positive all right so as long as we're clear about the sign of d from the beginning
i know which solution i need to choose and it's 1.06 meters of compression for those of you who
are expected to solve quadratic equations like this at least partly by hand i'll go ahead and
show that in our remaining space and i already arranged all these terms in descending order so
their coefficients are a b and c for the quadratic formula so i'd start with negative b that's 10.96
plus or minus the square root of b squared
minus 4 times a times c which has a double negative in it
so be careful with that all divided by 2 times a which was 25. so if i'm showing my work the long
way on this i'm going to simplify everything in the square root and take the square root
of it this gives me 10.96 plus or minus 42.00 all divided by 50. i can already tell the plus is the
one that gives me a positive answer so that's all i'm going to write down and i get my 1.06 meters
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