Question

calculate work done to pull a spring​

Answer 1

Work done to pull a spring is

W = -kx² { where , k = spring constant , x = distance pulled }

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Answer 2

To derive:-

Formula for work done to pull a spring (by an operator) $(\sf W.)$

Answer:-

Let the initial position of the spring be $\sf x = x_i,$ and the final position be $\sf x = x_f.$

From Hooke's law,

$\sf Spring\;force = F_s = - kx$

But, since we have to find work done by the operator,

$\sf Force_{(Man)} = - (-kx) = kx \quad \quad \dots (1)$

$\sf dW = F\:dx$

$\longrightarrow \displaystyle \sf \int_{0}^{W} dW = \int_{x_i}^{x_f} F\:dx$

$\longrightarrow \displaystyle \sf W = \int_{x_i}^{x_f} kx\:dx\quad [From\:(1)]$

$\longrightarrow \displaystyle \sf W = \bigg[k\; \; \dfrac{x^{(1+1)}}{1+1}\bigg]_{x_i}^{x_f}$

$\longrightarrow \displaystyle \sf W = \bigg[\dfrac{1}{2}kx^2\bigg]_{x_i}^{x_f}$

$\longrightarrow \underline{\underline{\sf{\green{W = \dfrac{1}{2}k\big(x_f { }^2 - x_i { }^2 \big) }}}}$