Determine the elastic potential energy stored in an elastic spring of natural length five meters and modulus of elasticity 13.4 newtons, which was compressed to a length of four meters.
Well, in this question, what we’re looking to find is elastic potential energy. And to find elastic potential energy, we have a formula. And that formula is that the elastic potential energy is equal to a half 𝐾𝑥 squared. Where 𝐾 is the spring constant, and that’s measured in newtons per meter. And 𝑥 is the magnitude of displacement, which is measured in m. So we can say that displacement is the amount that the spring is compressed or stretched. So it’s extended beyond its natural length.
Well, we can work out 𝑥 from the values that we’ve been given. And that’s because we’ve got five, which is the natural length of the spring, minus four, and that’s because four was the compressed length of the spring, which gives us one. So we know that 𝑥 is equal to one meter. Because we know the displacement away from the natural length of our spring is one meter. So this is great because we have 𝑥. But we need to ask ourselves, what’s the spring constant? Well, we can work out the spring constant using our modulus of elasticity and the natural length of both together. Well, we can find 𝐾, our spring constant, by dividing 𝐸, which is our modulus of elasticity, by our natural length.
And we can check this by looking at the units because we have 𝐸, which is our modulus of elasticity. And that’s measured in newtons. We’ve got 𝐿𝑜, which is our natural length, which is measured in meters. So we’ve got newtons divided by meters, which would give us newtons per meter, which is what the spring constant is measured in. So, therefore, if we substitute in our values, we can calculate 𝐾 is equal to 13.4 divided by five. So now, if we work this out, we get 13.4 divided by five is equal to 2.68 newton meters. So great, we’ve now found the spring constant. So we’ve got all the information that we need to work out the elastic potential energy.
So then we can now work out the elastic potential energy stored in our spring. And that’s gonna be equal to a half multiplied by 2.68 multiplied by one squared. And that’s the value of our spring constant multiplied by the value of 𝑥, so our displacement squared. So this is gonna give us 2.68 over two. And we get that because we had a half multiplied by 2.68 which is the same as 2.68 over two. Then we just multiplied the whole thing by one squared, which is just one.
So then, if we calculate this and find the value, we get 1.34 joules. So we can say that the elastic potential energy stored in the elastic spring of natural length five meters and modulus of elasticity 13.4 which was compressed to a length of four meters, is 1.34 joules.