Final answer:
To find the speed of the crate at t = 3.5 s, we first need to find the time needed to begin lifting the crate, which is approximately √7 seconds. Then, using the net force on the crate, the mass of the crate, and Newton's second law, we can find the acceleration. Finally, using the equation for speed, we can calculate the speed of the crate at t = 3.5 s, which is approximately 1.12 ft/s.
Step-by-step explanation:
To determine the speed of the crate at t = 3.5 s, we first need to find the time needed to begin lifting the crate. The force F pulling on the cable at A is given by F = 30 + t^2 lb. We can find the time needed to begin lifting the crate by setting F equal to the weight of the crate and solving for t: 30 + t^2 = 37. Rearranging the equation, we have t^2 = 7, and taking the square root of both sides, we get t = √7. Therefore, it takes approximately √7 seconds for the crate to begin lifting.
Next, we can find the speed of the crate at t = 3.5 s. From t = 0 to t = √7, the crate is on the ground and not moving. From t = √7 to t = 3.5, the force F is pulling the crate upward. The net force on the crate is the difference between the force F and the weight of the crate. The net force is given by F_net = F - weight = (30 + t^2) - 37. Plugging in t = 3.5, we have F_net = (30 + 3.5^2) - 37 = 49 - 37 = 12 lb. Using Newton's second law (F_net = ma) and solving for acceleration, we have acceleration = F_net / mass = 12 lb / 37 lb = 0.32 ft/s^2. Finally, we can find the speed of the crate at t = 3.5 s using the equation: speed = initial speed + (acceleration * time). Since the initial speed is 0 (the crate was at rest), we have speed = 0 + (0.32 * 3.5) = 1.12 ft/s. Therefore, the speed of the crate at t = 3.5 s is approximately 1.12 ft/s.