Final answer:
The tension in the rope when the 60.0-kg gymnast climbs at a constant speed is 588 N, and when he accelerates upwards at 1.50 m/s², the tension is 678 N. These calculations are based on the equilibrium of forces and Newton's second law of motion. Hence, option (a) is correct.
Step-by-step explanation:
In the scenario where 60.0-kg gymnast Kevin climbs a rope, we need to calculate the tension in the rope for two cases: when he climbs at a constant speed and when he accelerates upward. The tension in the rope under different conditions can be determined using 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 (F = ma).
(a) When Kevin climbs at a constant speed, his acceleration is zero, and the only force acting on him is gravity, which equals his weight. Therefore, the tension (T) in the rope must be equal to his weight to keep him moving upwards at a constant speed.
The weight is calculated as W = mg, where g is the acceleration due to gravity (9.8 m/s²).
So, T = mg = (60.0 kg)(9.8 m/s²) = 588 N.
(b) If Kevin accelerates upward at 1.50 m/s², the net force acting on him is not only his weight but also the force required for his acceleration, which is upward.
So, the total tension in the rope is now the combination of his weight and the force due to his acceleration:
T = mg + ma = m(g + a) = (60.0 kg)(9.8 m/s² + 1.50 m/s²) = (60.0 kg)(11.3 m/s²) = 678 N.