A spring with a constant of 16 newtons per metre has 98 joules of energy stored in it when it is extended. How far is the spring extended?
Alright, in this scenario, we have some spring. And let’s say that this is the spring at its equilibrium length. That is the length when it’s neither stretched nor compressed. Then, what happens is the spring, which is originally at its equilibrium length, is extended past that length. As a result of that extension, the spring gains the capacity to do work. That is, it stores energy within itself.
The problem statement tells us that 98 joules of energy are stored in this extended spring. And not only that, we’re also told how hard it is. That is how much force is needed in order to extend or compress the spring by one metre of length. And as we’re told, this value is the spring constant. Now if we call the distance the spring is extended beyond equilibrium 𝑑, then that’s the value that we want to solve for.
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. The potential energy stored in a spring, sometimes called the elastic potential energy, is equal to one-half the spring’s constant 𝑘 multiplied by the spring’s displacement, either being stretched or compressed from equilibrium, 𝑥 squared.
Now in our case, we can write out a slightly modified form of this equation. We can say that the elastic potential energy, or spring potential energy, is equal to one-half the spring constant 𝑘 multiplied by 𝑑 squared, where 𝑑 is the distance that the spring has been extended from equilibrium. We can start off solving for 𝑑 by rearranging this equation algebraically so that we have 𝑑 on one side by itself.
To do that, we can multiply both sides of the equation by two divided by 𝑘, the spring constant. Then, looking on the right-hand side, we see that that two cancels with the factor of one-half, and the 𝑘s cancel out as well. Next, to solve for 𝑑, we’ll take the square root of both sides of this equation. And taking this operation on the right-hand side cancels out with the square term of the 𝑑, leaving us with the equation the distance the spring is extended 𝑑 is equal to the square root of two times the potential energy stored in the spring divided by the spring constant.
Since we’re given that spring, or elastic, potential energy in the problem statement as well as the spring constant 𝑘, we can substitute in those values now. 𝑑 is equal to the square root of two times 98 joules divided by 16 newtons per metre. 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. That’s how far the spring has been extended.