Un cubo de masa m = 4,00 kg está unido a un resorte de constante k = 600 N/m como se muestra en la figura. El bloque se jala hasta una posición xi = 5,00 cm a la derecha del equilibrio y se suelta del reposo. Encuentra la velocidad que tiene el bloque al pasar por el equilibrio si a) la superficie horizontal no tiene fricción y b) el coeficiente de fricción entre el bloque y la superficie es µk=0,350

a)   v = 0.612 m / s , b) v = 0.179 m / s

Explanation:

We can solve this problem using the concepts of conservation of mechanical energy

starting point. Before releasing the cube

          Em₀ = = ½ k x²

final point. At the equilibrium point of the spring

          = K = ½ m v²

a) in this case there is no friction, so the energy is conserved

          Emo = Em_{f}

          ½ k x² = ½ m v²

          v = √k/m    x

let's reduce the magnitudes to the SI system

   x = 5.00 cm = 0.0500 m

let's calculate

         v = √600/4    0.05

         v = 0.612 m / s

b) In this part we are told that there is a friction, for which we use the work energy theorem

            W = ΔEm

friction force work is

           W = fr. x = fr x cos θ

the friction force opposes the movement, so the angle is 180º and the cosine is -1

the formula for friction force is

         fr = μ N

let's write Newton's second law on the vertical axis

         N - W = 0

         N = W

we substitute the friction force

         fr = μ m g

of the work energy equation

       - fr x = Em_{f} - Em₀

        - μ m g x = ½ m v² - ½ k x²

          v² = (½ k x² - μ m g x) 2 / m

          v = √(k / m x² - 2 μ g x)

          v = √ (600/4  0.05² - 2 0.350 9.8 0.05)

          v = 0.179 m / s

as we know our atmosphere or air have different levels. every level has other gasses, and every gas has another permeability. when the light comes from lower permeability to higher permeability, it changes its direction. so that means that when the light comes from the moon to the earth, it changes it's direction many more times. it basically means that the shape of the light part of the moon is changed on the earth.

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