Final answer:
The tension in the rope for a 4.5 slug gymnast climbing at constant speed is 144.783225 pounds. If the gymnast accelerates upward at 4.8 ft/s², the tension increases to 166.415625 pounds.
Step-by-step explanation:
The question involves a 4.5 slug gymnast climbing a rope and deals with calculating the tension in the rope under two differing conditions - constant speed and upward acceleration. To find these tensions, we apply principles of Newton's second law of motion. Given that a slug is a unit of mass in the Imperial system (where 1 slug = 32.17405 lb·ft/s·s), we convert the gymnast's mass to a weight force in pounds since tension will have the same units as force in this scenario.
Part a: Constant Speed
When the gymnast climbs at constant speed, the net force is zero because there is no acceleration (Newton's first law). The only forces acting on the gymnast are gravity (downward) and the tension in the rope (upward). These two forces must be equal and opposite, so the tension will be equal to the weight of the gymnast:
Tension at constant speed (T1) = weight of the gymnast = 4.5 slugs * 32.17405 lb·ft/s·s (acceleration due to gravity) = 144.783225 pounds
Part b: Upward Acceleration
When the gymnast accelerates upward at a rate of 4.8 ft/s², the tension must not only support the weight but also provide the extra force required for the acceleration. Using Newton's second law (F = ma), we determine the additional force needed and add it to the weight:
Tension with upward acceleration (T2) = weight of the gymnast + (mass * acceleration)
= 144.783225 pounds + (4.5 slugs * 4.8 ft/s²) = 144.783225 pounds + 21.6324 pounds = 166.415625 pounds