Question

To find the center of the lamina, we can use the center of mass formula. The x-coordinate of the center of mass, denoted as x-bar, is given by:

a) x-bar = ∫x dm / ∫dm
b) x-bar = ∫x dm / m
c) x-bar = ∫x dm / ∫x dx
d) x-bar = ∫x dm / ∫x dm

Answer 1

The x-coordinate of the center of mass of a lamina can be found using the formula x-bar = ∫x dm / ∫dm, where dm represents the mass element and x represents the position along the x-axis. By expressing dm as λ(dx), the formula becomes x-bar = ∫x λ(dx) / ∫ λ(dx).

Step-by-step explanation:

The x-coordinate of the center of mass, denoted as x-bar, is given by the formula x-bar = ∫x dm / ∫dm. This formula represents the average position of the mass along the x-axis. To use this formula, we need to integrate the position variable x with respect to the mass element dm. In this case, we can express dm as λ(dx), where λ is the linear mass density of the rod and dx is the infinitesimally small length element along the rod. After integrating, we have x-bar = ∫x λ(dx) / ∫ λ(dx), where we integrate from the initial position x = 0 to the final position x = L.