To solve this, we'll start with the key concepts:

1. The gymnast's weight:

The weight $W$ of an object is the force of gravity acting on it and is given by:

$W = m \cdot g$

Where:

• $m$ is the mass of the gymnast (50.0 kg),
• $g$ is the acceleration due to gravity (typically $9.8 \, \text{m/s}^2$).

Thus:

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

So, the gymnast's weight is 490 N.

2. The tension at the top of the rope:

In this case, since the rope has negligible mass and the gymnast is stationary (in equilibrium), the tension $T$ at the top of the rope must be equal to the weight of the gymnast to balance the force of gravity.

Therefore, the tension at the top of the rope is also:

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

Summary:

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

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:

1. How would the tension change if the gymnast climbed halfway up the rope?
2. What happens to the tension if the gymnast starts accelerating upwards?
3. How would the tension differ if the rope had significant mass?
4. What would be the gymnast's weight on the Moon where gravity is weaker?
5. How does the force exerted by the gymnast on the ground compare if she were standing?

Tip: Remember that tension is a force transmitted through a rope, string, or cable, and it adjusts to balance the forces in static or dynamic equilibrium.