Final answer:
The tension in the rope when the 64.0 kg box is at rest or moving up at a constant speed is 627.2 N. If the box is accelerating upwards at 5.20 m/s², the tension in the rope increases to 960.0 N to compensate for the acceleration.
Step-by-step explanation:
To calculate the tension in the rope in different scenarios when a box is hanging from it, we use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (Fnet = m × a). The weight of an object (w) is given by w = m × g, where m is the mass of the object and g is the acceleration due to gravity (9.80 m/s²). When a box is at rest or moving at a constant velocity, its acceleration is zero, and the tension in the rope equals the weight of the box. If the box is accelerating, the tension must compensate for both the weight of the box and the force required to accelerate it.
1. When the 64.0 kg box is at rest, the tension in the rope (T) is simply the weight of the box. Therefore, T = m × g = 64.0 kg × 9.80 m/s² = 627.2 N.
2. When the box moves upward at a steady 5.10 m/s, there is no acceleration, and the tension is still equal to the weight. T = 627.2 N.
3. When the box has a vertical velocity (vy = 4.70 m/s) and is speeding up at 5.20 m/s², the tension is the sum of the weight of the box and the force due to acceleration. T = m × g + m × a = 64.0 kg × 9.80 m/s² + 64.0 kg × 5.20 m/s² = 627.2 N + 332.8 N = 960.0 N.