An ideal spring has the spring constant k = 414 N/m. Calculate the distance this spring must be stretched from its equilibrium position for 22 J of work to be done in meters.
E =kx²/2,
x =sqrt(2E/k)
To calculate the distance that the spring must be stretched, we need to use the formula for the potential energy stored in a spring:
Potential Energy (PE) = (1/2) * k * x^2
Where:
- PE is the potential energy stored in the spring
- k is the spring constant
- x is the displacement or stretch (distance) of the spring from its equilibrium position
We are given that the work done on the spring is 22 J. We know that work is equal to the change in potential energy of the spring. In this case, the potential energy is increasing since work is done on the spring.
So, we can equate the work done on the spring to the change in potential energy:
Work = Change in Potential Energy
22 J = PE - 0
Since the spring is initially at its equilibrium position, the initial potential energy is 0 J.
Therefore, we have:
22 J = PE
Now, we can substitute the formula for potential energy and solve for x:
22 J = (1/2) * k * x^2
Rearranging the equation:
44 J = k * x^2
Substituting the value of the spring constant (k = 414 N/m):
44 J = 414 N/m * x^2
Dividing both sides of the equation by 414 N/m:
0.1063 m^2 = x^2
Taking the square root of both sides to solve for x:
x = √(0.1063 m^2)
x ≈ 0.326 m
Therefore, the distance this spring must be stretched from its equilibrium position for 22 J of work to be done is approximately 0.326 meters.