A position-dependent force F=3 x^2-2 x+7 acts on a body of mass 7 kg and displaces it from x=0

step 1 In this question we have given the position dependent force which is acting on a body whose mass

A position-dependent force F=3 x^2-2 x+7 acts on a body of...

A position-dependent force $F=3 x^{2}-2 x+7$ acts on a body of mass $7 \mathrm{~kg}$ and displaces it from $x=0 \mathrm{~m}$ to $x=5 \mathrm{~m}$. The work done on the body is $x^{\prime}$ joule. If both $F$ and $x$ are measured in SI units, the value of $x^{\prime}$ is (a) 135 (b) 235 (c) 335 (d) 935

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Key Concepts

Work Done by a Force

Work is defined as the transfer of energy by a force acting over a displacement. In physics, when the force is not constant but varies with position, the work done is calculated as the definite integral of the force function over the path of motion. This concept is crucial as it links the applied force and the resulting energy change in the system.

Position-dependent Force

A position-dependent force is one whose magnitude and/or direction varies as a function of position. Such forces require methods from calculus for analysis because their effect on an object cannot be accurately described by a simple multiplication of force and displacement. Instead, one must integrate the force over the path taken to determine the total work done.

Definite Integration

Definite integration is a fundamental calculus tool used to compute the net accumulation of a quantity, such as work, over an interval. In the context of variable forces, it allows for the summing of infinitesimal contributions of work (dW = F dx) over the displacement, providing a rigorous means to quantify work even when the force function is complex.

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