Final answer:
The tension in the vine, when Jill swings at the bottom of her path, is calculated by combining her weight with the centripetal force needed for the circular motion; it is approximately 651.4 N.
Step-by-step explanation:
To determine the tension in the vine when Jill is swinging, we can invoke the principles of physics. When Jill is at the bottom of the swing, where the vine is vertical, the tension (T) in the vine is the sum of the gravitational force (Jill's weight) and the centripetal force required to keep her moving in a circular path.
Jill's weight (W) is calculated using the equation W = mg, where m is Jill's mass (61 kg) and g is the acceleration due to gravity (9.81 m/s²). The centripetal force (Fc) needed for circular motion is given by Fc = mv²/r, where v is the tangential velocity (2.4 m/s) and r is the radius of the circle, which in this case is the length of the vine (6.5 m).
Combining these two forces gives T = W + Fc. Using the values given:
- W = 61 kg × 9.81 m/s² = 598.1 N
- Fc = (61 kg) × (2.4 m/s)² / (6.5 m) = 53.3 N
- T = 598.1 N + 53.3 N = 651.4 N
So, the tension in the vine when it is vertical and Jill is moving at 2.4 m/s would be approximately 651.4 N.