Thin rod nonuniform linear mass density

In summary, the moment of inertia of a thin rod with nonuniform density λ(x)=ae^-bx is given by -e^(bL)a[L^2/b + 2L/b^2 + 2/b^3] + 2a/b^3, where a is the linear density and b is the inverse of length. This solution is correct as long as the units for a and b are mass/length and 1/length, respectively.
  • #1
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Homework Statement



A thin rod of total length L has a nonuniform density given by λ(x)=ae^-bx where x is distance from one end of the rod. What is the moment of inertia of the rod for rotations around the end at x=0. Measuring from x=0 to x=L

Homework Equations


λ(x)=ae^-bx
Inertia=I=∫dmr^2
dm=λ(x)dx
r=x
I=mr^2


The Attempt at a Solution


my answer is -e^(bL)a[L^2/b + 2L/b^2 + 2/b^3] + 2a/b^3

I do not think this is correct because my units are not in mass*length^2

 
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  • #2
It looks correct. a and b mus have dimensions. a is linear density, [a]=[mass]/[length] and =1/[length].

ehild
 

1. What is a thin rod nonuniform linear mass density?

A thin rod nonuniform linear mass density refers to a physical object that has a varying mass per unit length along its length. This means that different sections of the rod have different masses, leading to a nonuniform distribution of mass along its length.

2. How is thin rod nonuniform linear mass density measured?

Thin rod nonuniform linear mass density is typically measured by dividing the mass of the object by its length. This gives the mass per unit length, which can then be compared to other objects with the same dimensions to determine the nonuniformity of the mass distribution.

3. What factors can affect the nonuniformity of a thin rod's linear mass density?

The nonuniformity of a thin rod's linear mass density can be affected by factors such as the material of the rod, its shape, and any external forces acting on it. These factors can cause variations in the mass distribution along the length of the rod.

4. How does thin rod nonuniform linear mass density impact the object's properties?

The nonuniform distribution of mass along a thin rod can impact its properties, such as its strength, stiffness, and ability to resist bending. Objects with a more uniform distribution of mass tend to have more predictable and consistent properties.

5. What are some real-life examples of thin rod nonuniform linear mass density?

Some real-life examples of thin rod nonuniform linear mass density include a fishing rod, a flagpole, and a bridge support beam. These objects have varying mass distributions along their lengths, which can affect their functionality and structural integrity.

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