Question

A spring of spring constant \( 200 \, {N/m} \) is initially stretched by \( 10 \) cm from the unstretched position. The work to be done to stretch the spring further by another \( 10 \) cm is:

• A. \( 3 \) J
• B. \( 6 \) J
• C. \( 9 \) J
• D. \( 12 \) J

Hint:

The work done in stretching a spring from \( x_1 \) to \( x_2 \) is calculated using \( W = \frac{1}{2} k x_2^2 - \frac{1}{2} k x_1^2 \). The energy stored in a spring follows Hooke’s Law.

Answer 1

The Correct Option is A

Step 1: Work done in stretching a spring
The work done in stretching a spring from an initial extension \( x_1 \) to a final extension \( x_2 \) is given by: \[ W = \frac{1}{2} k x_2^2 - \frac{1}{2} k x_1^2. \] Given: - Spring constant, \( k = 200 \) N/m,
- Initial extension, \( x_1 = 10 \) cm = \( 0.1 \) m,
- Final extension, \( x_2 = 20 \) cm = \( 0.2 \) m.
Step 2: Calculating the work done
Substituting the values: \[ W = \frac{1}{2} \times 200 \times (0.2)^2 - \frac{1}{2} \times 200 \times (0.1)^2. \] \[ W = \frac{1}{2} \times 200 \times 0.04 - \frac{1}{2} \times 200 \times 0.01. \] \[ W = 100 \times 0.04 - 100 \times 0.01. \] \[ W = 4 - 1 = 3 { J}. \] Step 3: Conclusion
Thus, the work required to stretch the spring further by another 10 cm is: \[ 3 { J}. \]