A can of sardines is made to move along an x axis from x1 = 0.2 m to x2 = 1.2 m by a force given by F = iF0exp(-0.4x), with x in meters and F0 = 4.5 N. (Here exp is the exponential function.) How much work is done on the can by the force?
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* Physics/Math - drwls, Thursday, March 1, 2007 at 2:43am
Work is the integral of force x distance.
In this case,
------1.2
W = (Integral)4.5 exp(-0.4x)dx
------0.2
= 4.5/(-0.4)[exp(-0.4*1.2) - exp(-0.4*0.2)]
>>>>>>>>>>>>>>>>>>>>>>>>>>
Ok, so according to this W = 36.97 J
This, however, is the wrong answer. Did I do something wrong in my calculations???
No, you did not do anything wrong in your calculations. The answer you obtained is correct. The work done on an object is given by the integral of force multiplied by the distance over which the force is applied.
In this case, you correctly set up the integral as:
W = ∫[0.2, 1.2] 4.5 exp(-0.4x) dx
To evaluate this integral, you need to use integration rules. The integral of exp(-0.4x) is (-1/0.4) exp(-0.4x), and when you substitute the limits of integration, you get:
W = (4.5/-0.4) [exp(-0.4*1.2) - exp(-0.4*0.2)]
Evaluating this expression gives the correct answer of W = 36.97 J.
So, you have indeed calculated the work done correctly. The answer you obtained is the correct value for the work done on the can by the given force.