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A balloon has 5 g of air. A small hole i...

A balloon has 5 g of air. A small hole is pierced into it. The air escapes at a uniform rate with a velocity of 4 cm/s. If the balloon shrinks completely in 2.5 s, then the average force acting on the balloon is

A

2 dyne

B

2 N

C

8 dyne

D

8 N

Text Solution

AI Generated Solution

The correct Answer is:
To find the average force acting on the balloon as the air escapes, we can follow these steps: ### Step 1: Calculate the mass of air escaping The total mass of air in the balloon is given as 5 g. Since the air escapes uniformly over 2.5 seconds, we first need to determine how much air escapes per second. **Hint:** To find the rate of mass loss, divide the total mass by the time taken for the air to escape. ### Step 2: Determine the rate of air escaping The rate of air escaping can be calculated as follows: - Total mass of air = 5 g - Time taken for air to escape = 2.5 s Rate of mass escaping (m_dot) = Total mass / Time = 5 g / 2.5 s = 2 g/s **Hint:** Remember to keep track of the units while calculating the rate. ### Step 3: Convert mass escaping to kg Since we need to work in SI units, we should convert grams to kilograms: - 1 g = 0.001 kg - Therefore, 2 g = 2 * 0.001 kg = 0.002 kg **Hint:** Always convert to SI units for consistency in calculations. ### Step 4: Calculate the momentum change The air escapes with a velocity of 4 cm/s. We need to convert this to meters per second: - 4 cm/s = 4/100 m/s = 0.04 m/s Now, we can calculate the momentum change per second (which is equal to the force): - Momentum change (p) = mass escaping * velocity = 0.002 kg * 0.04 m/s = 0.00008 kg·m/s **Hint:** Momentum is the product of mass and velocity, and the force can be derived from the change in momentum. ### Step 5: Calculate the average force Since the air escapes uniformly, the average force (F) acting on the balloon can be calculated as: - Average force (F) = rate of change of momentum = 0.00008 kg·m/s **Hint:** The average force can also be thought of as the rate of momentum change over time. ### Final Answer The average force acting on the balloon is **0.00008 N**.
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