The linear mass density, λ (x), for a one-dimensional object is plotted in the graph. The location of the center of mass for this object is rc from origin then \frac{9 r_{c}}{x

Engineering

Physics

Center of Mass

Summation of Vectors

Calculus

Question

The linear mass density, λ (x), for a one-dimensional object is plotted in the graph. The location of the center of mass for this object is r_c from origin then $\frac{9 r_{c}}{x_{0}}$ is

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For slaut part of curve

$m_{1} = \int_{0}^{x_{0}} (λ_{0} \frac{x}{x_{0}}) dr = \frac{λ_{0} x_{0}}{2}$

$x = \frac{1}{λ_{0} x_{0} / 2} \int_{0}^{x_{0}} x (\frac{λ_{0} x}{x_{0}}) dx = \frac{2 x_{0}}{3}$

for horizontal part

$m_{2} = λ_{0} x_{0}$

$x = \frac{3 x_{0}}{2}$

$∴= \frac{m_{1} \times x_{1 c} + m_{2} x_{2 c}}{m_{1} + m_{2}}$