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: $$3 { J}.$$