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 plane with a speed of 6 ft/s, determine how long it takes for the force to bring the cabinet to rest. F always acts parallel to the plane.

t₁ = 0.95 s

Explanation:

In this chaos we must use the definition of Newton's second law

F = m a = m dv / dt

dv = F dt / m

Let's replace and integrate, let's take the upward direction of the plane as positive, the force is positive

dv = ∫ (3 + 2t) dt / m

v = (3 t + 2 t² / 2) / m

Let's evaluate between the lower limit t = 0 v = - 6 ft / s (going down) to the upper limit t = t and v = 0

0 - (-6) = (3 (t - 0) + (t² - 0)) / m

t² + 3t - 6m = 0

Let's look for the mass

W = mg

m = W / g

m = 20/32

m = 0.625 slug

Let's solve the second degree equation

t² + 3t - 3.75 = 0

t = (-3 ± √ (32 + 4 1 3.75)) / 2

t = (-3 ± 4,899) / 2

t₁ = 0.95 s

t₂ = - 3.95 s

We take the positive time