An ideal spring of negligible mass is 12.00cm long when nothing is attached to it .when you hang a 3.15kg weight from it , you measure its length to be 13.40cm . If you want to store 10.0J of potential energy in this spring , what would be it's total length? Assume that it continues to obey Hooke's law .
the weight (force) of a 3.15 kg mass is ... 3.15 * g
find the spring constant (k) by dividing the force by the extension
... (3.15 * g) / (13.40 cm - 12.00 cm)
stored energy is ... 1/2 * k * x^2
... find x ... add it to 12
for Joules , the extension is in meters and the force is in Newtons
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.