Well, it sounds like this spring is taking its stretching exercises very seriously. 13.40 cm when a weight is attached? That's some spring in its step!
Now, if you want to store 10.0 J of potential energy in this spring, you need to find its new length. Let's use some mathematical clowning around to figure that out!
We know that the potential energy stored in a spring is equal to 1/2 * k * x^2, where k is the spring constant and x is the displacement from the equilibrium position.
Since the spring obeys Hooke's Law, we can say that k = F/x, where F is the force applied and x is the displacement.
In this case, the weight of 3.15 kg is being hung from the spring. So, the force F can be calculated as F = m * g, where m is the mass and g is the acceleration due to gravity.
Now, let's calculate F:
F = 3.15 kg * 9.8 m/s^2 = 30.87 N (approximately)
Using this force, we can calculate the spring constant k:
k = F/x = 30.87 N / (0.1340 m - 0.1200 m) = 258.92 N/m (approximately)
Okay, now let's solve for the displacement x when we want a potential energy of 10.0 J:
10.0 J = 1/2 * k * x^2
Simplifying the equation, we have:
20.0 N/m * x^2 = 10.0 J
Dividing both sides by 20.0 N/m:
x^2 = 0.5 J / (20.0 N/m) = 0.025 m
Taking the square root of both sides, we find:
x = 0.1581 m (approximately)
To find the total length, we just need to add the displacement to the original length:
Total length = 12.00 cm + 0.1581 m = 14.41 cm
So, to store 10.0 J of potential energy in this spring, it would have a total length of approximately 14.41 cm.
Remember, this is just a playful way to solve the problem, but it should give you the right answer in the end!