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CBSE Class 11-science Answered

Please explain the work done by a variable force.
Asked by kandappan | 09 Aug, 2019, 06:07: AM
answered-by-expert Expert Answer
In real life Applied force on an object will not be uniform and constant with respect to time and distance.
Variation in magnitude and direction with respect to time or distance may be possible. 
Let us consider an one-dimensional case, hence variation in magnitude is considered.
Fig. 1 is a plot of a varying force in one dimension. If the displacement Δx is small,
we can take the force F (x) as approximately constant and the work done is then,   ΔW =F (x) Δx
 
This is shown in Fig. 1. Adding successive rectangular areas in Fig. 1,  we get the total work done as
begin mathsize 12px style W space equals space sum from x subscript f to x subscript i of F left parenthesis x right parenthesis capital delta x space end style
where the summation is from the initial position xi to the final position xf. If the displacements are allowed to approach
zero, then the number of terms in the sum increases without limit, but the sum approaches a definite value
equal to the area under the curve as shown in Fig. 2.
Then the work done is  begin mathsize 14px style W space equals space limit as capital delta x rightwards arrow 0 of sum from x subscript i to x subscript f of F left parenthesis x right parenthesis capital delta x space equals space integral subscript x subscript i end subscript superscript x subscript f end superscript F left parenthesis x right parenthesis d x end style

where ‘lim’ stands for the limit of the sum when Δx tends to zero.
Thus, for a varying force the work done can be expressed as a definite integral of force over displacement
Answered by Thiyagarajan K | 09 Aug, 2019, 01:19: PM
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