Question

Find the center of mass of a rod of length L whose mass density changes from one end to the other quadratically. That is, if the rod is laid out along the x-axis with one end at the origin and the other end at x = L, the density is given by rho(x) = rho0 + (rho1 − rho0) x L 2 , where rho0 and rho1 are constant values. (Use the following as necessary: L, rho0, and rho1.)

Answer 1

To find the center of mass of a rod with changing density, divide the rod into small segments and calculate the mass of each segment. Integrate to find the x-coordinate of the center of mass.

Step-by-step explanation:

To find the center of mass of a rod with changing density, we can use the concept of a weighted average. We divide the rod into small segments, each with a mass dm. The mass of each segment can be calculated using the density function rho(x) = rho0 + (rho1 - rho0)(x/L)^2. The center of mass of each segment is located at x, and we can integrate to find the overall center of mass.

The formula for the x-coordinate of the center of mass is given by:

xcm = (1/M) * ∫(x * dm) = (1/M) * ∫(x * rho(x) * dx) = (1/M) * ∫(x * (rho0 + (rho1 - rho0)(x/L)^2) * dx), where M is the total mass of the rod.

By evaluating this integral, we can find the x-coordinate of the center of mass of the rod.

Answer 2

`x_(cm)` = ¾ L (ρ₀ + ρ₀) / (2ρ₀ + ρ₁)

Step-by-step explanation:

The center of mass of a body is defined as

Rcm =
`(1)/(M)` ∫ r dm

Where the blacks indicate vectors, Rcm is the position of the center of mass, M the total mass of the body, ‘r’ the position with respect to an origin

The density of a body is the relationship between two of its magnitudes that change constantly or vary in a known way, for this case we have

ρ = m / x

That can also be written in the form

ρ = dm / dx

dm = ρ dx

The expression they give us is

ρ₀ = ρ₀ + (ρ₁ -ρ₀) (x/L)²

dm =( ρ₀ + (ρ₁ -ρ₀) (x/L)² )dx

Replace in the center of mass equation and integrate, from the initial point x = 0 to the upper limit X = L . Since the whole system is on the x-axis, change the variable r by x (r --- x)

`x_(cm)` = M⁻¹ ∫ x [ρ₀ + ( ρ₁-ρ₀) (x/L)²] dx

`x_(cm)` = M⁻¹ [∫ ρ₀ x dx + I (ρ₁-ρ₀) / L² x³ dx]

`x_(cm)` = M⁻¹ [ρ₀ x²/ + (ρ₁ -ρ₀) / L² x⁴/4]

`x_(cm)` = M⁻¹ [ρ₀/2 (L2-0) + (ρ₀₁ -ρ₀) /4L² (L⁴-0)

`x_(cm)` = M⁻¹ [ρ₀ L²/2 + (ρ₁ -ρ₀)/4 L²]

`x_(cm)` = M⁻¹ L² [ρ₀/4 + ρ₁/4]

`x_(cm)`= M⁻¹ L²/4 ( ρ₀ +ρ₁)

The only parameter that we don't know explicitly is the total mass, but we can look for their relationship using the concept of density

M = ∫ dm = ∫ ρ dx

M = ∫ [ρ₀ + (ρ₁ -ρ₀)/L² x²] dx

We integrate and evaluate between the limits of integration x = 0 and x = L

M = ρ₀ x + (ρ₁ -ρ₀)/L² x³/3

M = ρ₀ L + (ρ₁ -ρ₀)/3L² L³

M = ρ₀ L + (ρ₁-ρ₀)/3 L

M = L (ρ₀ + ρ₁/3 -ρ₀/3)

M = L (2/3 ρ₀ + ρ₁/3)

M = L/3 (2ρ₀ + ρ₁)

Let's replace and simplify in the center of mass equation we have found

Xcm = L²/4 [ρ₀ + ρ₁] / [L/3 (2ρ₀ + ρ₀)/L]

Xcm = ¾ L (ρ₀ + ρ₀) / (2ρ₀ + ρ₁)