A bucket full of water weighs 5 kg, it is pulled from a well 20 m deep. There is a small hole in the bucket through which water leaks at a constant rate. If it is observed that for every meter the bucket loses 0.2 kg mass of water, then the total work done in pulling the bucket up from the well is `[g = 10ms^(-2)`]

Options

A. 600 J
B. 400 J
C. 100 J
D. 500 J

Answer

To solve the problem of calculating the total work done in pulling the bucket up from a well, we will follow these steps:

### Step 1: Understand the Problem
We have a bucket full of water weighing 5 kg, and it is being pulled up from a depth of 20 m. The bucket loses water at a rate of 0.2 kg for every meter it is pulled up.

### Step 2: Determine the Mass of the Bucket at Different Heights
As the bucket is pulled up, the mass of the water in the bucket decreases. The mass of the water left in the bucket at a height \( x \) (where \( x \) is the height from the bottom of the well) can be expressed as:
\[ m(x) = 5 \, \text{kg} - 0.2 \, \text{kg/m} \times x \]

### Step 3: Calculate the Weight of the Bucket at Height \( x \)
The weight of the bucket at height \( x \) is given by:
\[ W(x) = m(x) \cdot g = (5 - 0.2x) \cdot 10 \]
where \( g = 10 \, \text{m/s}^2 \).

### Step 4: Calculate the Work Done in Pulling the Bucket
The work done \( dW \) in pulling the bucket up a small distance \( dx \) is equal to the weight of the bucket at height \( x \) times the distance \( dx \):
\[ dW = W(x) \cdot dx = (50 - 2x) \, dx \]

### Step 5: Integrate to Find Total Work Done
To find the total work done \( W \) in pulling the bucket from the bottom of the well (0 m) to the top (20 m), we integrate \( dW \) from 0 to 20:
\[ W = \int_0^{20} (50 - 2x) \, dx \]

### Step 6: Perform the Integration
Calculating the integral:
\[ W = \int_0^{20} 50 \, dx - \int_0^{20} 2x \, dx \]
Calculating each part:
1. \(\int_0^{20} 50 \, dx = 50x \Big|_0^{20} = 50 \times 20 - 50 \times 0 = 1000\)
2. \(\int_0^{20} 2x \, dx = x^2 \Big|_0^{20} = 20^2 - 0^2 = 400\)

Thus,
\[ W = 1000 - 400 = 600 \, \text{J} \]

### Final Answer
The total work done in pulling the bucket up from the well is \( 600 \, \text{J} \).

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