Video Transcript
The amount of force it takes to compress a spring can be approximated using the
formula F equals 𝑘𝑥, where F is the force in newtons, 𝑘 is the spring constant in
newtons per millimeter, and 𝑥 is the displacement or change in the length of the
spring in millimeters. The given table shows the spring constants for seven different springs. Two springs, of types E and F, respectively, have a natural length of 100 millimeters
when no force is applied to them. A force of 40 N is applied to each of the springs in order to compress them. What is the difference in the length of the springs when subjected to the force of 40
N?
Before we can find the difference in the lengths of the springs after they are
compressed, we first need to find out how much each of the springs would be
compressed under the force of 40 N. We’re considering a type E spring and a type F spring. We’ve been told that F equals 𝑘 times 𝑥. The force is equal to the spring constant times the displacement.
But since we’re looking for the difference in lengths, we’re looking for the
displacement of E and F. We know the force applied, and we know the 𝑘 constant for each spring. If we divide both sides of this equation by 𝑘, we’ll see that 𝑥, the displacement,
is equal to the force divided by the spring constant. For springs E and F, we need to calculate the displacement by dividing the force by
the constant.
The force F for both springs is 40 newtons. And to find their spring constant, we need to check the table. Type E has a spring constant of 10 newtons per millimeter. So, we plug that in for 𝑘. And type F has a spring constant of 25 newtons per millimeter. 40 divided by 10 equals four. Remember that our 𝑥, our displacement, is measured in millimeters. Under a force of 40 newtons, the type E spring would change in length by four
millimeters. 40 divided by 25 is 1.6. The change in length of spring of type E would be 1.6 millimeters.
Let’s imagine two springs, one of type E and one of type F, that both start out at
the same length, 100 millimeters. When zero newtons are applied, when no force is applied, they’re the same length. When 40 newtons are applied to type E, it’s displaced by four millimeters. Then, when 40 newtons is applied to a spring type F, it’s displaced by 1.6
millimeters.
We find the difference in lengths of the springs by taking four millimeters and
subtracting 1.6 millimeters, which will give us 2.4 millimeters. This means that under a force of 40 newtons, the type F spring will be 2.4
millimeters longer than the type E spring and leaves our final answer 2.4
millimeters.
