00:01
Alright, hello, in this question we're told we have a spring with a spring constant and the mass given, and we're told we compress it to a density of 2 centimeters, and then we release it from rest, so v0 is equal to 0, and we're asked in part a, there's a force of friction which has a constant value of 3 .8 newtons, and we're asked where does the maximum speed of this occur? so i'm assuming that this is the compressed distance here, so we compress it some distance initially of delta x0, and then we know because we have a frictional force here, that it's going to reach its maximum speed before it gets back to that equilibrium point.
00:41
If there was no friction, it would have vmax here at its equilibrium point.
00:46
However, because there's friction, we'll assume that it's not going to reach the maximum speed after that, it's probably going to reach it at some intermediate point, i'll call it x1, and at that point we're going to have vmax.
01:00
So we want to use conservation of energy here, so we're going to have that our initial energy and our final energy are equal.
01:06
Initially, before we release this, we just have spring potential, right? there's no other sources of energy when it's not moving, and friction hasn't done anything.
01:15
At the end, we're going to potentially have some spring potential, we're going to have our kinetic energy, and that's going to be our maximum kinetic energy, and then some of this energy here is going to have to overcome friction.
01:28
So some of that initial energy is going to be lost to friction, so we're going to add the work done from friction here, and we're going to assume that this is just a positive value, we just want the quantity.
01:38
So i know that the force of friction opposes the motion, so technically the work of friction is negative, however, we just want to look at the positive quantity because we say, well, if 10 joules, for example, was lost in friction, that came from our initial energy.
01:53
So let's go ahead and expand these out, we have 1 half k delta x naught squared, that equals our final spring potential, 1 half k, we're at position x1 squared, and then we have a maximum speed of 1 half m v max squared.
02:08
And then we're going to add the work done by friction, which is going to be the force of friction times the distance we travel.
02:12
Well, we're traveling from this point over here, which is delta x naught, to this point here, x1, so we're going to have a travel distance of our initial distance minus wherever we end up with our maximum speed.
02:26
So now we look at this equation and say, well, we don't know what x1 is, and we don't know what v max is.
02:32
But we can say that if we don't look at our maximum speed, if we just say that we're at some point x1, and we're going to have some speed v, we can write this as v as a function of our position, because we have our independent variable here.
02:47
And then we can say that, well, with this, if we have v as a function of position, we can then go ahead and optimize that, take the maximum of it.
02:55
So i'm going to solve this for v, and this v is going to be a function of what we're calling x1.
03:01
And if we do that, we'll end up with this function here.
03:03
I then want to take the derivative of this and find the maximum.
03:06
So to take the derivative of this, well, i'm going to have 1 half times everything on the inside to the negative 1 half.
03:12
And then i have to take the derivative of the inside.
03:15
This is with respect to the variable x1.
03:18
So that portion will look like this...