The force required to stretch a Hooke's-law spring varies from 0 N to 76.4 N as we stretch the spring by moving one end 7.46 cm from its unstressed position.
Find the work done in stretching the spring. Answer in units of J.
To find the work done in stretching the spring, we need to integrate the force with respect to the displacement.
Since Hooke's law states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position, we can use the equation:
F = k * x
where F is the force, k is the spring constant, and x is the displacement.
In this case, we are given the maximum force (F = 76.4 N) and the displacement (x = 7.46 cm = 0.0746 m).
So the equation becomes:
76.4 N = k * 0.0746 m
Solving for k, we get:
k = 76.4 N / 0.0746 m ≈ 1024.06 N/m
Now we can find the work done by integrating the force over the displacement:
Work = ∫F dx
Work = ∫(k * x) dx
Work = k * ∫x dx
Since k is a constant, we can take it out of the integral:
Work = k * ∫x dx
Using the limits of integration from 0 to 0.0746 m, the work becomes:
Work = k * ∫[0 to 0.0746] x dx
Work = k * [x^2 / 2] from 0 to 0.0746
Work = k * (0.0746^2 / 2 - 0^2 / 2)
Work = k * (0.0746^2 / 2)
Work ≈ 1024.06 N/m * (0.0746^2 / 2)
Work ≈ 2.461 J
Therefore, the work done in stretching the spring is approximately 2.461 J.