Find the location of the center of mass for a one-dimensional rod of length L and of linear density ë(x) = cx, where c is a constant. (Hint: You will need to calculate the mass in terms of c and L. Use any variable or symbol stated above as necessary.)
X =
total mass*cm=INT density*x*dx=int cx2 cx= 1/3 c x^3= 1/3 cL^3
but total mass= INT denstiy dx= 1/2 cL^2
cm= 2/3 L measured from the low mass end.
To find the location of the center of mass for a one-dimensional rod with variable linear density, we need to integrate the product of the position and density over the length of the rod.
Given that the linear density of the rod is represented by ë(x) = cx, where c is a constant, we can express the mass of an infinitesimally small element dx as dm = ë(x)dx = cxdx.
We know that the total mass of the rod, M, is equal to the integral of the mass density over the entire length of the rod:
M = ∫ dm = ∫cxdx.
The length of the rod is given as L. Therefore, we need to evaluate the integral from x = 0 to x = L.
∫cxdx = c∫xdx = c[x^2/2] from 0 to L
= c(L^2/2 - 0^2/2)
= cL^2/2.
So, the total mass of the rod is M = cL^2/2.
Now, to find the position of the center of mass, we need to integrate the product of the position x and the mass density ë(x) over the length of the rod, and then divide by the total mass M:
X = (1/M) * ∫xë(x)dx.
Substituting ë(x) = cx, we have:
X = (1/(cL^2/2)) * ∫x(cx)dx.
Simplifying, we get:
X = (2/cL^2) * ∫x^2dx.
Evaluating the integral, we have:
X = (2/cL^2) * [x^3/3] from 0 to L
= (2/cL^2) * (L^3/3 - 0^3/3)
= (2L^3/3cL^2)
= 2L/3c.
Therefore, the location of the center of mass for a one-dimensional rod with linear density ë(x) = cx, where c is a constant, is at X = 2L/3c.