An empty balloon weighs `W_(1)`. If air equal in weight to `W` is pumped into the balloon, the weight of the balloon becomes `W_(2)`. Suppose that the density of air inside and outside the balloon is the same. Then

Options

A. `W_2=W_1`
B. `W_2=W_1+W`
C. `W_2ltW_1+W`
D. `W_2gtW_1`

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Conditions
- The weight of the empty balloon is denoted as \( W_1 \).
- When air equal in weight to \( W \) is pumped into the balloon, the new weight of the balloon becomes \( W_2 \).

### Step 2: Define the New Volume
- When air is pumped into the balloon, the volume of the balloon changes. The volume \( V \) of the air inside the balloon can be expressed in terms of the mass of the air and its density:
\[ V = \frac{\text{mass}}{\text{density}} = \frac{W/g}{\rho} \]
where \( W \) is the weight of the air added, \( g \) is the acceleration due to gravity, and \( \rho \) is the density of air.

### Step 3: Calculate the Real Weight of the Balloon
- The real weight of the balloon after adding the air is:
\[ \text{Real Weight} = W_1 + W \]

### Step 4: Consider the Buoyancy Force
- The balloon is in the atmosphere, which exerts a buoyant force \( F_B \) on it. The buoyant force is equal to the weight of the air displaced by the volume of the balloon:
\[ F_B = \rho g V \]
- From the previous step, we know that \( V = \frac{W}{\rho g} \), so substituting this into the buoyant force equation gives:
\[ F_B = \rho g \left(\frac{W}{\rho g}\right) = W \]

### Step 5: Calculate the Apparent Weight of the Balloon
- The apparent weight \( W_2 \) of the balloon (the weight we measure) is given by:
\[ W_2 = \text{Real Weight} - F_B \]
Substituting the values we found:
\[ W_2 = (W_1 + W) - W \]
Simplifying this gives:
\[ W_2 = W_1 \]

### Conclusion
- Therefore, we conclude that:
\[ W_2 = W_1 \]
- This means the apparent weight of the balloon after pumping in air equal to weight \( W \) is the same as the weight of the empty balloon.

### Final Answer
The correct option is that \( W_1 = W_2 \).

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