# Center of mass problem

1. Dec 15, 2008

### nns91

1. The problem statement, all variables and given/known data

A baseball bat of length L has a peculiar linear density given by $$\lambda$$ = $$\lambda$$0 * (1+X2/L2)

Find the x coordinate of the center of mass in terms of L

2. Relevant equations

Mxcm= mr

3. The attempt at a solution

So I use integration

The integrand I have is x*$$\lambda$$ dx and substitue whatever on the right side of the $$\lambda$$ equation in. Then I just took normal integral.

However I got wrong answer. The right answer does not contain $$\lambda$$0 but mine does

Can you guys help me ??

2. Dec 15, 2008

### Dr.D

You need to also do an integration to find the total mass, M. This will have a factor lambda_o that will cancel the one you have on the right side.

3. Dec 15, 2008

### nns91

How do I integrate to find mass M ??

Do I plug in lamda formula*L for M or do I have to integrate lamda*L ??

4. Dec 15, 2008

### Dr.D

M = int(lambda * dx) will do the job where you take the integral over the length 0 to L.

5. Dec 15, 2008

### nns91

I got (L^2/2 +L^3/4)/(1+L^2/3)

But it's still not the answer in the book.

Did I do something wrong ??

6. Dec 15, 2008

### LowlyPion

You've almost got it looks like to me.
But the numerator should have been multiplied by x

yielding as an integrand x + x3/L2

that gives

x2/2 + x4/4 | from 0 to L or ... 3L2/4

The total Mass looks integrated a little off.

Shouldn't that be (L + L/3) = 4L/3 ?

Then dividing denominator into numerator

(3/4L2)/(4/3*L) = 9L/16

Last edited: Dec 15, 2008
7. Dec 16, 2008

### nns91

How can you get 3L^2/4 and 4L/3

I thought the integral of the total mass will yield L+ L^3/3 ???

8. Dec 16, 2008

### LowlyPion

The integrand for the volume of the mass is 1 +X2/L2 |evaluated between 0 and L

That yields the result X + X3/3L2 The 0 terms are of no account leaving L + L3/3L2 = L + L/3 = 4L/3

The integrand for the incremental moments is as I outlined previously.