A plane lamina with constant density $\rho ( x , y ) = 12$ occupies a region in the $x y$-plane bounded by a simple closed path $C$. Its moments of inertia about the axes are
$$I _ { x } = - \frac { \rho } { 3 } \int _ { C } y ^ { 3 } d x \text { and } l _ { y } = \frac { \rho } { 3 } \int _ { C } x ^ { 3 } d y \text {. }$$
Find the moments of inertia about the axes, if $C$ is a rectangle with vertices $( 0,0 ) , ( 4,0 )$, $( 4,5 )$ and $( 0,5 )$.
Select the correct answer.
A) $( 2,000 , - 1,280 )$
B) $( - 1,280,1,280 )$
C) $( - 2,000,1,280 )$
D) $( 2000,1280 )$
E) $( - 2,880,1,200 )$

Question

Quizplus · Accepted Answer

The moments of inertia for a rectangle can be calculated using the given formulas. For a rectangle with vertices $(0,0), (4,0), (4,5), (0,5)$, the moments of inertia about the x-axis and y-axis are $I_x = \frac{\rho}{3} \int_0^4 \int_0^5 y^3 \, dy \, dx$ and $I_y = \frac{\rho}{3} \int_0^5 \int_0^4 x^3 \, dx \, dy$. Evaluating these integrals gives $I_x = 2000$ and $I_y = 1280$.

A Plane Lamina with Constant Density ρ(x,y)=12\rho ( x , y ) = 12ρ(x,y)=12

A Plane Lamina with Constant Density $\rho ( x , y ) = 12$