Answer to Question #180941 in Calculus for Larkin

Question #180941

A manufacturer wants to build a spring that takes a force 10 N (in negative direction) to compress it 0.2 m from the equilibrium position. The spring should be able to stretch 0.5 m from the equilibrium position.

As a mechanical engineer, you were asked to present how much work should be done to stretch the spring.The presentation shall include important components of the problem, complete and correct computations and a logical and organized explanation.

Hints:

First, create a force function F(x) by finding the spring constant k in F(x) = kx where F(x) is the force and x is the position from the equilibrium.

Then, calculate work ustng the concept of the definite integral.

If a variable force F(x) moves an object in a positive direction along the x-axis from point a to point b, then the work done on the object is W = ∫ ^b a F(x) dx


1
Expert's answer
2021-04-15T06:51:02-0400

Find the spring constant "k" . To do this, we will use the condition: to compress the spring by "x=0.2\\,\\,\\text{meters}" , it is necessary to apply a force of "F=10\\,\\,\\text{N}" .



"F=kx\\longrightarrow 10=k\\cdot0.2\\longrightarrow\\boxed{k=50\\,\\,\\frac{\\text{N}}{\\text{m}}}"

Now, let's find a general formula that will describe the amount of work that needs to be done in order to stretch the springs by "x" meters.



"W(x)=\\int\\limits_0^xF(x)dx=\\int\\limits_0^x kx'dx'=\\left.\\frac{k\\cdot\\left(x'\\right)^2}{2}\\right|_0^x=\\frac{kx^2}{2}"

In our case,



"k=50\\longrightarrow W(x)=\\frac{50x^2}{2}\\longrightarrow\\boxed{W(x)=25x^2}\\\\[0.3cm]\nW(0.5)=25\\cdot0.5^2=25\\cdot0.25=6.25\\,\\,\\text{N}"

The plot of energy versus elongation is as follows




ANSWER



"x=0.5\\longrightarrow W=6.25\\,\\,\\text{N}"



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS