Justin C. answered • 07/30/22
Chemical Engineer that LOVES teaching AP/College Calculus (5+ years)
Hooke's Law tells us that the Force of a spring can be found using the following equation:
Force = -k*x
where k is the spring constant and x is the distance of stretch (or compression) from equilibrium.
The simplified equation used to solve for Work is the following:
W = F*d
where F is force and d is distance over which the force is applied. However, this equation only works for situations where the force is constant.
Force is not constant across the distance that the spring is stretching because as the stretch distance increases, so does force. Using calculus, we can integrate the force equation with respect to distance to find work as follows:
W = integral(F*dx) = integral(k*x*dx)
The negative has been dropped as this will need to be determined based upon the direction of movement and the direction in which the force is being applied.
Integrating this function leaves us with the following equation:
W = 1/2*k*x^2
This should look familiar, as it is similar to the equation of the potential energy of a spring according to Hooke's Law. This would be the equation used if your course is not calculus-based. The previous step was just to explain how this formula is obtained. To solve for the work done, we will need to evaluate this equation at the point after the stretch and the point before the stretch and find the difference in the values.
As you can see, we will need a value for k. To find k, we can use this equation and the information given in the problem:
W = 1/2*k*x^2
10.5 ft*lb = 1/2*k*(0.5ft)^2
k = 84 lb/ft
Finally, evaluating our equation for potential energy at each of 6 inches and 6.5 inches:
PE = 1/2*k*x^2
PE2 = 1/2*(84 lb/ft)*(0.54166ft)^2 = 12.32 lb*ft
PE1 = 1/2*(84 lb/ft)*(0.5ft)^2 = 10.5 lb*ft
The difference between these two is as follows:
W = 12.32 - 10.5 = 1.82 lb*ft
Let me know if you have any questions!
