A spring of force constant K is first stretched by distance a from its natural length and then further by distance b. The work done in stretching the part b is

Options

A. `(1)/(2)Ka(a-b)`
B. `(1)/(2)Ka(a+b)`
C. `(1)/(2)Kb(a-b)`
D. `(1)/(2)Kb(2a+b)`

Answer

To solve the problem of calculating the work done in stretching a spring by an additional distance \( b \) after it has already been stretched by a distance \( a \), we can follow these steps:

### Step 1: Understand the work done in stretching a spring
The work done on a spring when it is stretched from its natural length to a distance \( x \) is given by the formula:
\[ W = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant.

### Step 2: Calculate the work done in stretching the spring to distance \( a + b \)
When the spring is stretched to a total distance of \( a + b \), the work done is:
\[ W_{total} = \frac{1}{2} k (a + b)^2 \]

### Step 3: Calculate the work done in stretching the spring to distance \( a \)
The work done in stretching the spring to distance \( a \) is:
\[ W_a = \frac{1}{2} k a^2 \]

### Step 4: Determine the work done in stretching the spring from \( a \) to \( a + b \)
The work done in stretching the spring from distance \( a \) to \( a + b \) (which is the work done in part \( b \)) can be calculated by subtracting the work done to stretch to \( a \) from the total work done to stretch to \( a + b \):
\[ W_b = W_{total} - W_a \]
Substituting the expressions we found:
\[ W_b = \frac{1}{2} k (a + b)^2 - \frac{1}{2} k a^2 \]

### Step 5: Simplify the expression
Now, we simplify \( W_b \):
\[ W_b = \frac{1}{2} k \left( (a + b)^2 - a^2 \right) \]
Expanding \( (a + b)^2 \):
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
Thus, substituting back:
\[ W_b = \frac{1}{2} k \left( a^2 + 2ab + b^2 - a^2 \right) \]
This simplifies to:
\[ W_b = \frac{1}{2} k (2ab + b^2) \]

### Step 6: Factor out the common terms
We can factor out \( b \) from the expression:
\[ W_b = \frac{1}{2} k b (2a + b) \]

### Final Answer
The work done in stretching the part \( b \) is:
\[ W_b = \frac{1}{2} k b (2a + b) \]

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