A thin rod of total length L has a nonuniform linear mass density given by lambd

Question

A thin rod of total length L has a nonuniform linear mass density given by lambda (x) = a(1+e^(-bx))
a and b is constant
1. What is total mass of the rod
2. What is the center of mass of the rod
3. What is moment of inertia of the rod for rotations around an axis perpendicular to the rod and passing through the end of the rod at x=0
4. suppose that the torque vector t = t n^ ( n hat ) acting perpendicular to the rod at the end of the rod at x=0. What is angular acceleration of the rod ?

I think for number 1-3 I need to intergrate something, but I didn't study calculus, it's just general physic but I don't know why my teacher give homework like this. And for number 4, I don't even know where to start.
Any help is much appreciate.

Explanation / Answer

Unfortunately you have to integrate, there is no other way.

Total mass is the integral of dx from 0 to L

Center of mass is Integral of x dx from 0 to L.

Moment of inertia is integral of x2 dx from 0 to L.

Moment of inertia (also called rotational inertia) uses the symbol I. The rotational analog of F = ma is = I, where tau is torque, I is rotational inertia and alpha is the angular acceleration you are looking for. So if you have I, they gave you torque so you can solve that equation for alpha.

Moment of inertia, unlike the other two, depends on the origin chosen but the integral I wrote works for the origin the problem has.

There isn't room here to show you how to integrate; sorry I can't help more. But that's what you've got here. You can actually set these problems up and maybe get some partial credit.

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