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.)

= ¾ L (ρ₀ + ρ₀) / (2ρ₀ + ρ₁)

Explanation:

The center of mass of a body is defined as

       Rcm =  ∫ 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)

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

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

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

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

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

      = M⁻¹ L² [ρ₀/4 + ρ₁/4]

     = 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)

the answer is to check all of them! they all apply well at least i think so!

explanation:

as the height from which the ruler was dropped increased, the speed of the ruler at ryan's hand increased.