Center of mass of baseball bat problem

[nns91]

Homework Statement

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

Homework Equations

Mx_cm= mr

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 ??

 

[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.

 

[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 ??

 

[Dr.D]

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

 

[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 ??

 

[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

 

[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 ?

 

[LowlyPion]

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

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

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.