The 20-lb cabinet is subjected to the force F = (3 + 2t) lb, where t is in seconds. If the cabinet is initially moving down the

Answer:

t₁ = 0.95 s

Explanation:

In this scenario, we apply Newton's second law, defined as

      F = m a = m dv / dt

      dv = F dt / m

Let's substitute and integrate, considering the upward direction of the plane as positive and that the force is also positive.

       dv = ∫ (3 + 2t) dt / m

       v = (3 t + 2 t²/ 2) /m

Evaluating between the lower limit where t = 0 and v = -6 ft/s (descending) to the upper boundary t = t and v = 0 gives:

       0 - (-6) = (3 (t- 0) + (t² -0)) / m

       t² + 3t -6m = 0

Next, we seek to find the mass:

      W = mg

      m = W / g

      m = 20/32

      m = 0.625 slug

Now let's solve the quadratic equation:

     t² + 3t -3.75 = 0

     t = (-3 ± √ (32 + 4 * 1 * 3.75)) / 2

     t = (-3 ± 4.899) / 2

From this, we have:

     t₁ = 0.95 s

     t₂ = -3.95 s

We select the positive time value.