To solve this problem, we need to first find the spring constant, also known as the spring stiffness, using Hooke's law equation. Then we can use the potential energy formula for a spring to find the total length.
Let's break down the problem step-by-step:
Step 1: Find the spring constant (k)
Hooke's law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, it can be written as:
F = -kx
Where:
F is the force applied to the spring
k is the spring constant
x is the displacement of the spring from its equilibrium position
In this problem, we know that when there is no weight attached, the spring's length is 12.00 cm, and when a 3.15 kg weight is attached, the length becomes 13.40 cm.
To find the displacement (x), we need to subtract the equilibrium length (length without any weight) from the length with the weight:
Displacement (x) = 13.40 cm - 12.00 cm = 1.40 cm
Next, we can use Hooke's law to find the spring constant (k):
F = -kx
k = -F / x
The force acting on the spring can be calculated using Newton's second law:
Force (F) = mass (m) × acceleration due to gravity (g) = 3.15 kg × 9.8 m/s^2
Substituting the values:
k = -(3.15 kg × 9.8 m/s^2) / (1.40 cm)
Note: We need to convert the displacement (1.40 cm) to meters (m) to match the units of force.
Step 2: Find the total length of the spring to store 10.0 J of potential energy
The potential energy stored in a spring can be calculated using the formula:
Potential Energy (PE) = (1/2)kx^2
We want to store 10.0 J of potential energy in the spring, so we can rearrange the formula and solve for x:
x = sqrt((2 × PE) / k)
Substituting the values:
x = sqrt((2 × 10.0 J) / k)
Finally, we can find the total length by adding the displacement (x) to the equilibrium length:
Total Length = Equilibrium Length + Displacement (x)
Substituting the values, we can calculate the total length of the spring.
Note: Make sure to convert the total length back to centimeters if needed.
I hope this helps! Let me know if you have any further questions.