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Question

The linear mass density of a rod of length 5 m varies with distance from its end fixed at the origin as represented in the graph shown below:

Then find the position of centre of mass of the rod w.r.t origin.
  1. 2516 m
  2. 59 m
  3. 259 m
  4. 516 m


Solution

The correct option is C 259 m
From the graph of linear mass density of rod, writing it in y=mx+c format:

λ=(5+x) kg/m

Let us consider a small element of length dx and mass dm, which is at a distance x from the origin. 


Let xCM be the position of COM of rod, so by applying basic formula and integrating with proper limits:
xCM=x dmdm...(i)

λ=dmdx=5+x
dm=(5+x)dx....(ii)

Substituting dm in Eq (i) and setting the values of limits as x=0 to x=5:

xCM=50x(5+x)dx50(5+x)dx=50(5x+x2)dx50(5+x)dx
=[5x22+x33]50[5x+x22]50
=[5(5)22+(5)33][5(5)+(5)22]
xCM=259 m

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