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 find the work done in stretching the spring from a distance \( a \) to a distance \( a + b \), we can follow these steps:

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

### Step 2: Calculate Initial and Final Potential Energy
1. **Initial Stretch**: When the spring is stretched by a distance \( a \), the potential energy \( U_1 \) is:
\[ U_1 = \frac{1}{2} k a^2 \]

2. **Final Stretch**: When the spring is further stretched by a distance \( b \), the total stretch becomes \( a + b \). The potential energy \( U_2 \) at this point is:
\[ U_2 = \frac{1}{2} k (a + b)^2 \]

### Step 3: Expand the Final Potential Energy Expression
Now, we can expand \( U_2 \):
\[ U_2 = \frac{1}{2} k (a + b)^2 = \frac{1}{2} k (a^2 + 2ab + b^2) \]

### Step 4: Calculate the Work Done
The work done \( W \) in stretching the spring from \( a \) to \( a + b \) is the difference between the final and initial potential energies:
\[ W = U_2 - U_1 \]
Substituting the expressions for \( U_2 \) and \( U_1 \):
\[ W = \left(\frac{1}{2} k (a^2 + 2ab + b^2)\right) - \left(\frac{1}{2} k a^2\right) \]

### Step 5: Simplify the Expression
Now, simplifying the expression for work done:
\[ W = \frac{1}{2} k (a^2 + 2ab + b^2 - a^2) \]
\[ W = \frac{1}{2} k (2ab + b^2) \]
\[ W = \frac{1}{2} k b (2a + b) \]

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

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