Question

A rod of length L is placed along the X-axis between x=0 and x=L. The linear density (mass/length) rho of the rod varies with the distance x from the origin as rhoj=a+bx. a. Find the SI units of a and b. b. Find the mass of the rod in terms of a,b, and L.

Answer 1

Answer:

See the answer in the image.

Explanation:

See the answer in the image provided.

Answer 2

The centre of mass of the rod lies at [(3aL+2bL^2)/(6a+3aL),0,0]

Explanation:

The COM of the element has co-ordinates (x, 0, 0). Therefore, x-coordinate of COM of the rod will be

xCOM = ∫L-0 xdm / ∫L-0 dm = ∫L-0 (x) (a+bx) dx  / ∫L - 0 (a+bx)bx

= ax^2 / 2 + bx^3 / 3 ÷ [ax+bx^2 / 2]L-0

or xCOM = 3aL + 2bL^2 / 6a + 3bL

The y-coordinate of COM of the rod is

Y COM=∫ydm / ∫dm = 0 (asy=0)

Similarly,

z COM = 0

So, the centre of mass of the rod lies at [(3aL+2bL^2)/(6a+3aL),0,0]