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)

-first use of geothermal energy occurred more than 10,000 years ago in north america by american paleo-indians

-the first industrial use of geothermal energy began near pisa, italy in late 18th century

-in 1904, italian scientist piero ginori conti invented the first geothermal electric power plant in which steam was used to generate the power

-with the above experiment, the first geothermal plant in usa started in 1922 with a capacity of 250 kilowatts

-however, in 1946 first ground-source geothermal heat pump installed at commonwealth building in portland, oregon

-during the 1960’s, pacific gas and electric began operation of first large scale geothermal power plant in san francisco, producing 11 megawatts

-in 1973, when oil crisis began many countries began looking for renewable energy sources and by 1980’s geothermal heat pumps (ghp) started gaining popularity in order to reduce heating and cooling costs

-as effect of climate change started showing results, governments of various countries joined hands to fight against it, for which kyoto protocol was signed in japan in 1997, laid out emission targets for rich countries and required that they transfer funds and technology to developing countries, 184 countries have ratified it

hope this