Find Mass & Center of Mass for a Rod of Length 38.5 cm

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SUMMARY

The discussion focuses on calculating the mass and center of mass for a rod of length 38.5 cm with a variable linear density defined by the equation λ = 50.0 g/m + 16.5 g/m². To find the mass, users must integrate the linear density over the length of the rod, resulting in the total mass. For determining the center of mass, participants are advised to apply the general expression for center of mass, which involves further integration of the density function multiplied by the distance from the reference point.

PREREQUISITES
  • Understanding of linear density and its mathematical representation
  • Knowledge of integration techniques in calculus
  • Familiarity with the concept of center of mass
  • Basic physics principles related to mass distribution
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  • Learn about the derivation of the center of mass formula
  • Explore applications of linear density in physics problems
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Homework Statement


A rod of length 38.5 cm has linear density (mass-per-length) given by the following equation, where x is the distance from one end.

λ=50.0g/m+16.5g/m^2

a. What is its mass in g?

b. How far from the x=0 end is its center of mass in m?

Homework Equations





The Attempt at a Solution



For part a., could I just divide by the mass?

Could someone please show me how to do this problem? I really don't understand what it's asking for.

Thank you very much
 
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For part A:

By definition,
mass = \lambda \cdot length

Since linear density isn't constant, you will need to integrate. But first you need an expression for how the mass of an infinitesimal length of the rod. Can you figure out what this is?

After you have an expression for how dm relates to dx, you can integrate both sides to get the total mass.For part B, start with the general expression for center of mass.
 
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