Final answer:
To find the spring constant, we can use Hooke's Law and calculate the displacements for two different masses hanging from the spring. Substituting the values into the equation F = -kx, we can solve for k and find the spring constant. The spring constant is approximately 18.31 N/m.
Step-by-step explanation:
To find the force constant, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. The equation is given by F = -kx, where F is the force, k is the spring constant, and x is the displacement. In this case, we have two sets of data: when a 0.25 kg mass hangs from the spring, the length is 0.225 m, and when a 1.975 kg mass hangs from it, the length is 0.75 m.
Using the data, we can calculate the displacements. For the 0.25 kg mass, the displacement is 0.225 m - 0 m = 0.225 m. For the 1.975 kg mass, the displacement is 0.75 m - 0 m = 0.75 m.
Now we can solve for the spring constant by using the equation F = -kx. For the 0.25 kg mass, the force is mg = 0.25 kg * 9.8 m/s^2 = 2.45 N. Substituting the values into the equation, we get -2.45 N = -k * 0.225 m. Solving for k gives us k = 2.45 N / 0.225 m = 10.89 N/m.
For the 1.975 kg mass, the force is mg = 1.975 kg * 9.8 m/s^2 = 19.31 N. Substituting the values into the equation, we get -19.31 N = -k * 0.75 m. Solving for k gives us k = 19.31 N / 0.75 m = 25.74 N/m.
Since we have two values for the spring constant, we can take the average to find the overall spring constant:
(10.89 N/m + 25.74 N/m) / 2 = 18.31 N/m.
Therefore, the spring constant k is approximately 18.31 N/m.