A rod has a linear density $\lambda$ = 3x,where x is the distance along the rod,which spans 0

Question

A rod has a linear density $\lambda$ = 3x,where x is the distance along the rod,which spans 0 < x < 1 m.What is the coordinate of the center of mass of the rod?
A)0.25 m
B)0.33 m
C)0.5 m
D)0.67 m
E)0.75 m

Quizplus · Accepted Answer

The center of mass $x_{cm}$ of a rod with a variable linear density $\lambda(x) = 3x$ can be found using the formula $x_{cm} = \frac{1}{M}\int_{0}^{1} x\lambda(x) dx$, where $M$ is the total mass of the rod. First, find the total mass $M = \int_{0}^{1} \lambda(x) dx = \int_{0}^{1} 3x dx = \frac{3}{2}x^2\Big|_0^1 = \frac{3}{2}$. Then, calculate the moment $ \int_{0}^{1} x\lambda(x) dx = \int_{0}^{1} 3x^2 dx = \frac{3}{3}x^3\Big|_0^1 = 1$. Thus, $x_{cm} = \frac{1}{\frac{3}{2}} = \frac{2}{3}$ m, which is 0.67 m.

A Rod Has a Linear Density λ\lambdaλ = 3x,where X Is the Distance Along the Rod,which Spans

A Rod Has a Linear Density $\lambda$ = 3x,where X Is the Distance Along the Rod,which Spans