A spring with a force of 5.2 N/m has a relaxed length of 2.45. When a mass is attached to the end of the spring and allowed to come to rest. the vertical length of the spring is 3.57m. Calculate the elastic potential energy stored in the spring.
To calculate the elastic potential energy stored in the spring, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
Hooke's Law equation: F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant, and
x is the displacement from the equilibrium position.
In this case, the force exerted by the spring is given as 5.2 N/m, and the displacement from the equilibrium position (the relaxed length) is 3.57 m - 2.45 m = 1.12 m.
Therefore, we can calculate the force exerted by the spring using Hooke's Law:
F = -kx
5.2 N/m = -k * 1.12 m
Solving for k, the spring constant:
k = -5.2 N/m / 1.12 m
k ≈ -4.64 N/m
Since the spring constant is negative in this case, it indicates that the spring is compressed.
The formula for elastic potential energy stored in a spring is:
PE = (1/2) * k * x^2
Where:
PE is the elastic potential energy, and
x is the displacement from the equilibrium position.
Now, substituting the values into the formula, we get:
PE = (1/2) * (-4.64 N/m) * (1.12 m)^2
PE = (1/2) * (-4.64 N/m) * 1.25 m^2
PE ≈ -2.9 J
The negative sign indicates that the elastic potential energy is stored in the spring, meaning that work needs to be done to compress or stretch the spring.