Question

In physics, Hooke’s law states that the length x of a uniform spring is a linear function of the force, y, applied to it. If we write y = a + kx, then the coe?cient k is called the spring constant. You are asked to determine the spring constant of a particular spring. This particular spring has an unstretched measured length of 6.1 inches (i.e., x = 6.1 when y = 0). You realize that because of experimental error, you want to take more than one measurement. Using forces of 2 pounds, 4 pounds, and 6 pounds applied to the spring, you measure the corresponding lengths to be 7.6 inches, 8.7 inches, and 10.4 inches, respectively. Find the best estimate for the spring constant of this spring?

Answer 1

To determine the spring constant of a particular spring, we can use Hooke's law. Hooke's law states that the force applied to a spring is directly proportional to the displacement or change in length of the spring. By using the measurements of force and length provided, we can calculate the spring constant.

Step-by-step explanation:

To determine the spring constant of a particular spring, we can use Hooke's law. Hooke's law states that the force applied to a spring is directly proportional to the displacement or change in length of the spring. The equation for Hooke's law is F = -kx, where F is the force, k is the spring constant, and x is the displacement.

In this case, we have three measurements of force and their corresponding lengths. We can use these measurements to calculate the spring constant. Let's use the measurement where y = 0, which corresponds to the length x = 6.1 inches. Plugging these values into the equation y = a + kx, we get 0 = a + 6.1k. Since y = 0, we can solve for a and get a = -6.1k.

Next, we can use the other measurements of force and length. For the force of 2 pounds, the length is 7.6 inches. Plugging these values into the equation, we get 2 = a + 7.6k. Substituting the value of a that we found earlier, we get 2 = -6.1k + 7.6k, which simplifies to 2 = 1.5k.

Similarly, for the force of 4 pounds and the force of 6 pounds, we can set up the corresponding equations and solve for k. Using these three equations, we can find the value of k that satisfies all of them and gives the best estimate for the spring constant.