To solve this problem, we need to understand the concept of work done in the context of blowing a bubble. When you blow a bubble, you are doing work against the surface tension of the liquid to increase the surface area of the bubble.
The work done (W) to blow a bubble is directly proportional to the change in the surface area of the bubble. For a spherical bubble, the surface area (A) is given by the formula:
A = 4πR²
where R is the radius of the bubble.
The work done to blow a bubble from a radius of R to a radius of 3R can be calculated by finding the difference in the surface area before and after the change in radius and then multiplying by the surface tension (γ) of the liquid:
Work done (W') = γ * (A_final - A_initial)
Let's calculate the initial and final surface areas:
A_initial = 4πR²
A_final = 4π(3R)² = 4π(9R²) = 36πR²
Now, let's find the difference in surface area:
ΔA = A_final - A_initial = 36πR² - 4πR² = 32πR²
The work done to increase the radius from R to 3R is:
W' = γ * ΔA = γ * 32πR²
Since we are not given the value of the surface tension (γ), we cannot calculate the exact work done. However, we can express the work done in terms of the initial work done (W) to blow the bubble from a radius of R to R. The initial work done is proportional to the initial surface area:
W = γ * 4πR²
Now, we can express the work done to increase the radius from R to 3R in terms of W:
W' = (γ * 32πR²) / (γ * 4πR²) * W
W' = (32/4) * W
W' = 8W
Therefore, the work done to increase the radius of the bubble from R to 3R is 8 times the work done to blow the bubble from a radius of R to R.
Hence, the correct answer is option b. 8W
To solve this problem, we need to understand the concept of work done in the context of blowing a bubble. When you blow a bubble, you are doing work against the surface tension of the liquid to increase the surface area of the bubble.
The work done (W) to blow a bubble is directly proportional to the change in the surface area of the bubble. For a spherical bubble, the surface area (A) is given by the formula:
A = 4πR²
where R is the radius of the bubble.
The work done to blow a bubble from a radius of R to a radius of 3R can be calculated by finding the difference in the surface area before and after the change in radius and then multiplying by the surface tension (γ) of the liquid:
Work done (W') = γ * (A_final - A_initial)
Let's calculate the initial and final surface areas:
A_initial = 4πR²
A_final = 4π(3R)² = 4π(9R²) = 36πR²
Now, let's find the difference in surface area:
ΔA = A_final - A_initial = 36πR² - 4πR² = 32πR²
The work done to increase the radius from R to 3R is:
W' = γ * ΔA = γ * 32πR²
Since we are not given the value of the surface tension (γ), we cannot calculate the exact work done. However, we can express the work done in terms of the initial work done (W) to blow the bubble from a radius of R to R. The initial work done is proportional to the initial surface area:
W = γ * 4πR²
Now, we can express the work done to increase the radius from R to 3R in terms of W:
W' = (γ * 32πR²) / (γ * 4πR²) * W
W' = (32/4) * W
W' = 8W
Therefore, the work done to increase the radius of the bubble from R to 3R is 8 times the work done to blow the bubble from a radius of R to R.
Hence, the correct answer is option b. 8W