What is the Minimum Size L for Uniform Density with Less Than 1% Error?

[xdrgnh]

Homework Statement

The linear mass density in a string is given by μ = μ0[1 + cos(x/R)] where R is a constant. If one averages this density over the large size L it becomes uniform: <μ> = μ0, where <…> means averaging. What is the minimum size L (in terms of R) such that the density can be considered uniform with an error less than 1% ?

Homework Equations

The Attempt at a Solution

So I intergrate with respect to dx over the range o to L then divide by L because I'm averaging and what I get is U+UR/L*sin(L/R). However this is my problem. The initial mass density makes no sense. When x=R*pi the density is zero. How can the density be zero on a freaking string. That makes no sense. Besides that I don't now what is meant by error. Should I equal the U+UR/L*sin(L/R) to .99U then solve?

 

[xdrgnh]

I appears in my title I messed up. I mean variable density.

[Moderator's note: thread title has been corrected by Redbelly98]

 

[Redbelly98]

The Attempt at a Solution

So I intergrate with respect to dx over the range o to L then divide by L because I'm averaging and what I get is U+UR/L*sin(L/R). However this is my problem. The initial mass density makes no sense. When x=R*pi the density is zero. How can the density be zero on a freaking string. That makes no sense.

You're right that it can't be zero on a real string. Better to think of it as negligibly small compared to the average, and calling it zero is an approximation.

Besides that I don't now what is meant by error. Should I equal the U+UR/L*sin(L/R) to .99U then solve?

Yes. Ideally, it should be solved twice, using both 1.01U and 0.99U.

 

[Chestermiller]

Don't forget that the maximum and minimum values that sin(L/R) can take on are +1 and -1.

Chet

 

[Nickie]

I would first clarify the definitions and assumptions being made in this scenario. It seems that the string has a variable linear mass density, which is described by the function μ = μ0[1 + cos(x/R)], where x is the position along the string and R is a constant. However, it is stated that when this density is averaged over a large size L, it becomes uniform with an average density of μ0.

Assuming that the string is initially non-uniform in density, it is not surprising that there may be a point where the density is zero, as long as it is still physically possible for the string to exist in this state. It would be important to clarify the physical meaning and implications of this point in the context of the problem.

To determine the minimum size L for the density to be considered uniform with an error less than 1%, I would first define what is meant by "error" in this context. Is it referring to the difference between the average density and the actual density at any given point? Or is it referring to the overall deviation of the density from the average? Once this is clarified, I would use the given function for the density and the definition of average to set up an equation and solve for L in terms of R. This would give a minimum value for L that satisfies the given criteria.