00:01
In this problem, we're reminded that we usually ignore the kinetic energy of the moving coils of a spring, but let's try to get a reasonable approximation of this.
00:11
So consider a spring of mass m, length l -not, and force constant k.
00:17
The work done to stretch or compress the spring by a distance l is 1 .5 kx squared, where x is l minus l -not, or the change in length.
00:27
Consider a spring ascribed to broth that has one end fixed and the other end moving with speed v.
00:35
Assume that the speed of points along the length of the spring varies linearly with the distance l from the fixed end and assume also that the mass of the spring is distributed uniformly along its length.
00:46
Okay, so we can draw a little schematic of a spring here.
00:53
I've used a coordinate x here, i guess they called it little.
00:56
But same thing and one end of this spring is fixed although we could actually say that this was v not and generalize this a bit but this case we just considered a one end of the spring is grounded the other end has a velocity v sub l and we have uniform density along the spring and it's we're gonna assume that the velocity varies linearly.
01:29
So at this point we have a velocity of zero here and here we have a velocity of v -l at x equals l and so our linear velocity changes is this line here.
01:42
So the velocity grows linearly to this point here.
01:47
Now what we need to do is we need to figure out how much kinetic energy is stored in this spring with this linear velocity change along its length.
02:00
So what we can do is again we note that the spring is uniform so we can determine a density per unit length or linear density so that's the mass of the spring divided by the length of the spring so that's the mass per unit length along the spring and it's a constant.
02:24
And so what we actually want to do what we're trying to look look for here is this integral, 1 half v squared dm over the whole body.
02:35
Okay, so that's the integral that we want to do because that's one half mv squared, but this is if the velocity is changing over the body, then we need to do an integral here.
02:49
And so what we can do is we say that the differential mass element here is the differential is the density times differential length so we can substitute in that dm equals row l d x and now we have a coordinate d x that we can integrate along so our our kinetic energy integral here reduces down to this which is an integral we can do easily so we're basically just adding up all of these little mass elements we're summing them up each one has a as a differential kinetic energy and we're gonna sum them all up and get the total kinetic energy over the length of the spring so we get one half the integral from zero to l row sub l v squared d x and we have we have an equation here for v as a function of x, and that is v .l over l times x.
04:01
So we plug that in and we square it.
04:05
And so now we can, now we have an integral as a function of x over dx, and we can work with that.
04:15
And a lot of stuff comes out.
04:19
So this, all this stuff here is constants, and we can pull it out.
04:22
And then we get x squared d x from zero to l and that's just l cubed over three okay um so and i've again substituted row in here as m over l and so we get one -half m vl squared over l cubed times l cubed over three the l cubes obviously cancel and so we get one -half m over three v time v l -l the velocity at the end of the spring squared.
04:56
And so what you can say here is that we basically have an effective mass of one -third that of the spring.
05:02
So if we wanted to, we could take a real spring that has mass.
05:07
Let's say we'll draw it a thick spring here that has mass.
05:12
And instead make it a thin spring here with no mass.
05:16
Again, these are all going to be fixed at this end for this formula to work...