Question

The spring is initially in unstretched condition. It is first stretched by a length '\(x\)' and the work done is \(W_1\). Then again it is stretched by further length '\(x\)' when the work done is \(W_2\). The value of \(W_2\) is given by

• A. \(W_1\)
• B. \(2W_1\)
• C. \(3W_1\)
• D. \(4W_1\)

Hint:

Work to stretch from 0 to x is half k x squared; from x to 2x subtract it from the work to 2x.

Answer 1

The Correct Option is C

Step 1: Work in Stretching a Spring:
The work to stretch a spring from natural length by \(s\) is \(\dfrac12ks^2\).

Step 2: First Stretch:
\(W_1=\dfrac12kx^2\).

Step 3: Second Stretch:
Total work to reach extension \(2x\): \(\dfrac12k(2x)^2=2kx^2\). So
\[ W_2=2kx^2-\frac12kx^2=\frac32kx^2=3W_1 \]

Step 4: Check the Options:
Option (A) \(W_1\) would be true for a constant force, but a spring force grows with extension. Option (B) \(2W_1\) and (D) \(4W_1\) match neither \(\tfrac32kx^2\) nor the difference found above.

Final Answer:
\(W_2=3W_1\), option (C). \[ \boxed{\text{(C) } 3W_1} \]