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Question

A thin rod lies along the x-axis from x=0 to x=2.7cm. The linear density of the rod, λ, is given by the equation: λ=1+ex and has units of g/cm.
Where is the Center of Mass of the rod?

A
At x=0.55cm
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B
At x=1.35cm
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C
At x=1.50cm
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D
At x=1.70cm
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E
At x=1.81cm
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Solution

The correct option is E At x=1.81cm
Consider a small element dx at a distance x from the origin.
The center of mass is defined as COM=2.70x(λdx)2.70λdx
Now, 2.70x(λdx)=2.70x(1+ex)dx=|x22|2.70+|xexex|2.70=3.64+40.1814.88+1=29.94 and
2.70λdx=2.70(1+ex)dx=|x+ex|2.70=2.7+14.881=16.58
Thus, COM=29.94/16.58=1.81cm
Hence, the center of mass is at x=1.81cm

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