WEBVTT
00:00:00.270 --> 00:00:06.810
A spring with a constant of 16 newtons per metre has 98 joules of energy stored in it when it is extended.
00:00:07.190 --> 00:00:09.020
How far is the spring extended?
00:00:09.520 --> 00:00:12.150
Alright, in this scenario, we have some spring.
00:00:12.210 --> 00:00:15.180
And letโs say that this is the spring at its equilibrium length.
00:00:15.440 --> 00:00:18.140
That is the length when itโs neither stretched nor compressed.
00:00:18.440 --> 00:00:23.980
Then, what happens is the spring, which is originally at its equilibrium length, is extended past that length.
00:00:24.280 --> 00:00:28.450
As a result of that extension, the spring gains the capacity to do work.
00:00:28.680 --> 00:00:30.790
That is, it stores energy within itself.
00:00:31.070 --> 00:00:35.960
The problem statement tells us that 98 joules of energy are stored in this extended spring.
00:00:36.200 --> 00:00:38.960
And not only that, weโre also told how hard it is.
00:00:39.000 --> 00:00:44.840
That is how much force is needed in order to extend or compress the spring by one metre of length.
00:00:45.250 --> 00:00:47.950
And as weโre told, this value is the spring constant.
00:00:48.360 --> 00:00:54.790
Now if we call the distance the spring is extended beyond equilibrium ๐, then thatโs the value that we want to solve for.
00:00:55.100 --> 00:01:02.540
That distance as well as the spring constant and the amount of energy stored in the spring are all connected to one another through a mathematical equation.
00:01:02.860 --> 00:01:17.130
The potential energy stored in a spring, sometimes called the elastic potential energy, is equal to one-half the springโs constant ๐ multiplied by the springโs displacement, either being stretched or compressed from equilibrium, ๐ฅ squared.
00:01:17.610 --> 00:01:21.380
Now in our case, we can write out a slightly modified form of this equation.
00:01:21.630 --> 00:01:33.380
We can say that the elastic potential energy, or spring potential energy, is equal to one-half the spring constant ๐ multiplied by ๐ squared, where ๐ is the distance that the spring has been extended from equilibrium.
00:01:33.720 --> 00:01:39.830
We can start off solving for ๐ by rearranging this equation algebraically so that we have ๐ on one side by itself.
00:01:40.210 --> 00:01:45.240
To do that, we can multiply both sides of the equation by two divided by ๐, the spring constant.
00:01:45.620 --> 00:01:51.810
Then, looking on the right-hand side, we see that that two cancels with the factor of one-half, and the ๐s cancel out as well.
00:01:52.140 --> 00:01:56.540
Next, to solve for ๐, weโll take the square root of both sides of this equation.
00:01:56.860 --> 00:02:10.500
And taking this operation on the right-hand side cancels out with the square term of the ๐, leaving us with the equation the distance the spring is extended ๐ is equal to the square root of two times the potential energy stored in the spring divided by the spring constant.
00:02:10.810 --> 00:02:19.010
Since weโre given that spring, or elastic, potential energy in the problem statement as well as the spring constant ๐, we can substitute in those values now.
00:02:19.430 --> 00:02:24.900
๐ is equal to the square root of two times 98 joules divided by 16 newtons per metre.
00:02:25.270 --> 00:02:33.220
When we enter this expression on the left-hand side of our equation on our calculator, to two significant figures, we find a result of 3.5 metres.
00:02:33.490 --> 00:02:35.840
Thatโs how far the spring has been extended.