Let's solve the problem step by step.

Given Data

1. Mass of empty bucket, $m_b = 5 \, \text{kg}$
2. Acceleration during lowering, $a = 5 \, \text{m/s}^2$
3. Force applied when raising the bucket, $F = 100 \, \text{N}$
4. Acceleration when raising, $a' = 2 \, \text{m/s}^2$
5. Gravitational acceleration, $g = 9.8 \, \text{m/s}^2$

We will find: (a) The force in the rope when lowering the bucket. (b) The capacity in liters of water added to the bucket.

Part (a): Force in the rope when lowering the bucket

The net force acting on the bucket while it is being lowered can be expressed as:

$T = m_b(g - a)$

Substitute the values:

$T = 5(9.8 - 5) = 5 \times 4.8 = 24 \, \text{N}$

Thus, the force in the rope when lowering the bucket is 24 N.

Part (b): Capacity of water added to the bucket

When the bucket is raised, the total mass is the combined mass of the bucket and water, $m = m_b + m_w$, where $m_w$ is the mass of the water.

The force equation for raising the bucket is:

$F = m(g + a')$

Substitute the known values:

$100 = (5 + m_w)(9.8 + 2)$

$100 = (5 + m_w)(11.8)$

Solve for $m_w$:

$5 + m_w = \frac{100}{11.8} \approx 8.47$

$m_w \approx 8.47 - 5 = 3.47 \, \text{kg}$

The mass of the water is approximately 3.47 kg.

To find the capacity in liters, use the fact that $1 \, \text{kg}$ of water corresponds to $1 \, \text{liter}$:

$\text{Capacity} = 3.47 \, \text{liters}$

Final Answers

1. Force in the rope when lowering the bucket: 24 N
2. Capacity of water added to the bucket: 3.47 liters

Do you want further clarification on any step or additional explanation?
Here are some related questions to expand your understanding:

1. What role does acceleration play in determining the tension in a rope?
2. How would the force change if the acceleration of raising the bucket was higher?
3. How can we calculate tension if the rope's weight isn't neglected?
4. What is the significance of gravitational acceleration in these calculations?
5. How could this scenario change if the bucket were submerged in water?

Tip: Always isolate forces acting on a system when using Newton's second law