Mechanics question involving spring and hookes law.

In summary, the conversation discusses finding the value of V using the relation Work Done = ∫F.dx = ∆K.E. The steps involved include integrating by parts and using the limits of x=4 and x=0. The final expression for V is given as V= 8[x.arctan(x/3)+3/2log(9/9+x^2)]. The value of λ is not specified and is left as a variable.
  • #1
ly667
7
0
My lamda value is 4.

What I have managed to get so far is:

dV/dx=-F

V=8/arctan(x/3)dx /=integrate sign!

integrate by parts

u=arctan(x/3), dv=dx
du=3/9+x^2dx, v=x

V=8/arctan(x/3)dx = 8[x.arctan(x/3)-/3x/9+x^2dx]

V= 8[x.arctan(x/3)-3/2log(9+x^2)+C]

Since V(0)=0

V= 8[x.arctan(x/3)-3/2log(9+x^2)+3/2log(9)]
V= 8[x.arctan(x/3)+3/2log(9/9+x^2)]


Not sure if this is correct, or where to go from here? Please help!
 

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  • #2
Hi ly667. It looks like you are using the relation
Work Done = ∫F.dx = ∆K.E.

and you will integrate between the limits of x=4 and x=0

I recommend that you get this written down at the start of the problem before you leap into the mechanics of problem solving, so that from the outset you can be confident of knowing precisely where you are headed. If you get into the habit of first laying down a good foundation, most of the time the solution will rise up almost Phoenix-like out of that foundation. :smile:

I'm not sure that your λ is 4, so let's keep it at λ.
V= 8[x.arctan(x/3)-3/2log(9+x^2)+C]
I don't quite get that. Can you do it again, this time keeping λ in the expression?
 

1. What is Hooke's Law?

Hooke's Law is a principle that states the force needed to extend or compress a spring is directly proportional to the distance the spring is extended or compressed.

2. How do you calculate the force of a spring?

The force of a spring can be calculated using the equation F = -kx, where F is the force, k is the spring constant, and x is the distance the spring is extended or compressed.

3. What is the relationship between the spring constant and the stiffness of a spring?

The spring constant is a measure of the stiffness of a spring. A higher spring constant indicates a stiffer spring, meaning it requires more force to extend or compress it compared to a spring with a lower spring constant.

4. Can Hooke's Law be applied to all types of springs?

Hooke's Law can be applied to most types of springs, as long as the spring remains within its elastic limit. However, some materials, such as rubber, may not follow Hooke's Law as they exhibit non-linear behavior.

5. How does temperature affect the behavior of a spring?

In general, an increase in temperature can cause a spring to become less stiff, resulting in a decrease in its spring constant. However, some materials, such as metals, may exhibit an increase in stiffness with increasing temperature, known as thermal hardening.

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