If one of the masses of the atwood's machine below is 3.2 kg, what should be the other mass so that the displacement of either mass during the first second following release is 0.23 m? assume a massless, frictionless pulley and a massless string

 m₂ = 2.91 kg

Explanation:

Let's analyze the exercise, ask us for the mass of the body we could find with Newton's second law and we need the acceleration that we can calculate with kinematics

Let's start looking for acceleration with kinematics

          y = vo t + ½ a t²

They indicate for the first second (t = 1 s) it descends y = 0.23 m

        y =  ½ a t²

        a = 2 y / t²

        a = 2 0.23 / 1²

        a = 0.46 m/s²

Let's look for the mass of the Atwood machine with Newton's second law, let's write the equations for each mass

        W₁- T = m₁ a

        T - W₂ = m₂ a

Let's add the two equations

      W₁ -W₂ = (m₁ + m₂) a

      m₁ g - m₂ g = m₁ a + m₂ a

      m₂ (a + g) = m₁ (g-a)

      m₂ = m₁ (g-a) / (g + a)

      m₂ = 3.2 (9.8 -0.46) / (9.8 + 0.46)

      m₂ = 3.2 9.34 / 10.26

     m₂ = 2.91 kg

the area between the graph and the t axis