A position dependent force F⃗=(7-2 x+3 x^2) N acts on a small abject of mass 2 k g to displace i

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A position dependent force F⃗=(7-2 x+3 x^2) N acts on a...

A position dependent force $\vec{F}=\left(7-2 x+3 x^{2}\right) N$ acts on a small abject of mass $2 k g$ to displace it from $x=0$ to $x=5 m$. The work done in joule is $\quad$ [CBSE PMT 1994]
(a) $70 J$
(b) $270 J$
(c) $35^{J}$
(d) $135 J$

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Key Concepts

Definite Integration

Definite integration is a mathematical technique used to accumulate quantities that vary continuously over an interval. In physics, it is commonly applied to compute work done by a variable force by integrating the force function with respect to displacement over a specific range.

Position-Dependent Forces

Position-dependent forces are forces whose magnitude or direction changes as a function of an object's location. Understanding these forces is essential in analyzing systems where the force is not uniform, allowing the use of calculus to determine quantities such as work or potential energy differences.

Work Done by a Variable Force

This concept refers to the calculation of work when the force acting on an object is not constant but changes with position. Instead of simply multiplying force by displacement, work is obtained by integrating the force function over the distance that the object moves, which effectively sums the contributions of force at every point along the path.

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