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Question

(B) The work done $W$ by a variable force $F$ acting on a particle moving from position $x_i$ to $x_f$ is given by the integral $W = \int_{x_i}^{x_f}

Vedclass · Accepted Answer

$\frac{1}{2}cx_1^2$

Vedclass · Answer

(B) The work done $W$ by a variable force $F$ acting on a particle moving from position $x_i$ to $x_f$ is given by the integral $W = \int_{x_i}^{x_f} F \, dx$.Given the force $F = cx$ and the limits of integration from $x = 0$ to $x = x_1$,we substitute these into the formula:$W = \int_{0}^{x_1} cx \, dx$Since $c$ is a constant,we can take it out of the integral:$W = c \int_{0}^{x_1} x \, dx$Using the power rule for integration,$\int x^n \, dx = \frac{x^{n+1}}{n+1}$,we get:$W = c \left[ \frac{x^2}{2} \right]_{0}^{x_1}$$W = c \left( \frac{x_1^2}{2} - \frac{0^2}{2} \right)$$W = \frac{1}{2} cx_1^2$

$A$ particle moves under the effect of a force $F = cx$ from $x = 0$ to $x = x_1$. The work done in the process is

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