To solve this, let's break it into parts:

1. The Gymnast's Weight

The weight of an object is given by the equation:

$W = m \cdot g$

Where:

• $W$ is the weight,
• $m = 50.0 \, \text{kg}$ is the mass of the gymnast,
• $g \approx 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.

Substituting the values:

$W = 50.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 490 \, \text{N}$

So, the gymnast's weight is 490 N (Newtons).

2. Tension at the Top of the Rope

The tension at the top of the rope is equal to the force needed to support the gymnast's weight since the rope is in static equilibrium (no movement). Therefore, the tension $T$ at the top of the rope is simply the same as the gymnast's weight:

$T = W = 490 \, \text{N}$

Thus, the tension at the top of the rope is 490 N.

Summary:

• The gymnast's weight is 490 N.
• The tension at the top of the rope is also 490 N.

Would you like further details on the concepts of weight and tension? Here are five related questions to explore the topic further:

1. How does the tension change if the gymnast starts moving upward?
2. What would the tension be if the rope had mass?
3. How does the tension vary at different points along the rope in this scenario?
4. If the gymnast's mass were doubled, how would her weight and the tension change?
5. How would the tension differ if the gymnast were on the moon, where gravity is weaker?

Tip: When solving for tension in situations involving ropes and static objects, always consider all forces acting on the system to determine equilibrium conditions.