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A particle is subjected to a variable force F(x)=6x N, where x is in metres. What is the work done in moving the particle from x=0 m to x=3 m?
54 J
9 J
18 J
27 J
27 J
The work done by a variable force is equal to the integral of the force function over the displacement interval. For the force F(x)=6x, the work done from x=0 to x=3 is calculated by evaluating the definite integral тИл03тАЛ6xdx, resulting in 27J.
The work done by a variable force is equal to the integral of the force function over the displacement interval. For the force F(x)=6x, the work done from x=0 to x=3 is calculated by evaluating the definite integral тИл03тАЛ6xdx, resulting in 27J.
Stretching a spring is like this; the further you pull it, the more force you need, and the total energy you put in is the cumulative effect of that increasing resistance.
WIF (Work is Integral of Force)
W=тИлx1тАЛx2тАЛтАЛF(x)dx тАФ Definition of work done by a variable force
F(x)=6x тАФ Given variable force function
тИлxndx=n+1xn+1тАЛ тАФ Power rule of integration
Work done (W) by a variable force F(x) along the x-axis is given by the area under the force-displacement graph. Mathematically, it is defined as the integral W=тИлx1тАЛx2тАЛтАЛF(x)dx. Integrating 6x yields 3x2, and applying the limits [0,3] gives 3(3)2тИТ3(0)2=27J.
Work is defined as the scalar product of force and displacement for constant forces, but integration is required for variable forces.
The area under a Force-Displacement (F-x) graph represents the work done.
The SI unit of work is the Joule (J), where 1J=1NтЛЕm.
Calculus provides a precise method to solve physics problems where force is not constant.
Calculating work done in compressing or stretching springs (Hooke's Law).
Analyzing energy changes in non-uniform gravitational or electric fields.
The force F(x)=6x implies a linear relationship where force increases proportionally with displacement.
Option A (54 J) is the result if one mistakenly integrates to 6x2. Option B (9 J) arises from failing to account for the coefficient in the integration. Option C (18 J) is a common calculation error.
D is correct тАФ The work done by the variable force F(x)=6x is the definite integral from 0 to 3, which yields 27J.
Always check if the force is constant or variable before applying W=Fd. If the force contains x, t, or v, use integration.
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