Mean radius of a cylinder

In summary, the problem is asking for the mean radius, \bar{r}, from the midpoint of a cylinder with radius a and height h to its boundary surface. The relevant equation is \bar{r} = (1/4pi)\int\int^ r sin\varthetad\varthetad\beta. The limits for beta should be 0 to 2pi for a cylinder, and for theta, 0 to pi. Expressing r in terms of theta can be done using cylindrical coordinates (s,\phi,z). The general formula for averaging a function over a surface \mathcal{S} can be used to solve the problem. The answer given by Attix's textbook is 11.
  • #1
kmoh111
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0

Homework Statement



I am working a homework problem that is trying to find the mean radius, [tex]\bar{r}[/tex] from the midpoint of a cylinder.

The problem states:
What is the mean radius, [tex]\bar{r}[/tex] from the midpoint of a cylinder of radius a and height h to its boundary surface? Evalute mean radius [tex]\bar{r}[/tex] for a = h/2 = 10 cm.

Homework Equations



The relavent equation is [tex]\bar{r}[/tex] = (1/4pi)[tex]\int[/tex][tex]\int^[/tex] r sin[tex]\vartheta[/tex]d[tex]\vartheta[/tex]d[tex]\beta[/tex]


The Attempt at a Solution


The problem and formula above is from Attix's textbook. In this case I think the limits for beta need to be 0 to 2pi for a cylinder.

I'm not sure what the limits for theta should be. I'm think it's 0 to pi.
I need to express r in terms of theta - but I'm not sure how.

Attix gives the answer as 11.32 cm.

Thanks in advance for any assistance.
 
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  • #2
I would use cylindrical coordinates [itex](s,\phi,z)[/itex] if I were you...the distance from the center of the cylinder (the origin) to a general point on the cylinder [itex](s,\phi,z)[/itex] is then [itex]\sqrt{s^2+z^}[/itex]...then all you need to do is average that over all three surfaces of the cylinder.

What is the general formula for averaging a function over a surface [itex]\mathcal{S}[/itex]?...Use that.
 

1. What is the meaning of mean radius of a cylinder?

The mean radius of a cylinder is the average distance from the center of the cylinder to its outer boundary. It is calculated by dividing the total radius of the cylinder by 2.

2. How is the mean radius of a cylinder different from the radius of a cylinder?

The radius of a cylinder refers to the distance from the center of the cylinder to its outer boundary at any given point. The mean radius, on the other hand, is the average of all these distances.

3. Why is the mean radius of a cylinder important?

The mean radius of a cylinder is an important parameter in many mathematical and engineering calculations. It is used to calculate the volume, surface area, and other properties of the cylinder.

4. How do you calculate the mean radius of a cylinder?

The mean radius of a cylinder can be calculated by measuring the radius at several points along the cylinder's length and taking the average of these measurements. Alternatively, it can be calculated by dividing the total radius by 2.

5. Can the mean radius of a cylinder change?

Yes, the mean radius of a cylinder can change if the radius at different points along the cylinder's length is not consistent. It can also change if the cylinder is deformed or damaged in some way.

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