Final answer:
The tension in the rope when the box is at rest on frictionless ice is 539 N.
Step-by-step explanation:
Tension in the rope must equal the weight of the supported mass, as we can prove using Newton's second law. If the 55.0 kg box is at rest on frictionless ice, then its acceleration is zero, and thus the net force on the box is zero. The only external force acting on the box is its weight, which is equal to the tension in the rope. Therefore, the tension in the rope is equal to the weight of the box, which can be calculated using the formula:
Tension = (mass) * (acceleration due to gravity)
Given that the mass of the box is 55.0 kg and the acceleration due to gravity is 9.8 m/s^2, the tension in the rope is:
Tension = (55.0 kg) * (9.8 m/s^2) = 539 N