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Line integral workdone12/18/2022 ![]() This is exactly the area one can get using line integral. When the area above the force curve is cut off, the remaining region gives us the area along the curve. Subsequently, we can calculate the area of the remaining sheet of paper and this gives the value of work done. So in the case of line integral, we can use the ‘paper’ analogy where one of the edges is traced along the path and along the perpendicular edge, force is marked. We will then mark the value of the force at every point (in the case of an object moving along x-axis) of x-axis on y-axis, and cut off the paper above the curve. Imagine a sheet of paper and assume one of the edges is X-axis and the perpendicular edge to be Y-axis. If you are given the co-ordinates of the complicated curve and the magnitude of force at every point on the curve, to measure the work done, we have to multiply force and the tiny displacements at every point on the curve. This cannot be done in a straightforward manner using integral calculus earlier discussed and one needs to invoke the concept of line integral. Then the work done by that force is given by integral of force multiplied by tiny displacement ‘ds’ (For any clarification on this, refer the previous article in which integral calculus is discussed in elaborate detail.)Ĭoming to the point of the article, let’s say the object is moved by the force not along x-axis but along a complicated curve in the Cartesian co-ordinate system. For example, when an object moved by a displacement-dependent-force along X-axis from origin to point A and the force at every point on X-axis is given as a function of ‘x’. We are familiar with single-variable integrals of the form a b f ( x ) d x, a b f ( x ) d x, where the domain of integration is an interval a, b. compute the work done by the force field on a particle that moves along the curve C that is the counterclockwise quarter unit circle in the first quadrant. 6.2.4 Describe the flux and circulation of a vector field. ![]() 15.2.042 - Find work done by the force field on a. 6.2.3 Use a line integral to compute the work done in moving an object along a curve in a vector field. Let’s say the force isn’t constant in the whole process and varies with respect to the amount of displacement. AKPotW: Using line integrals to find the work done by a force along a curve. Work done is a measure of energy spent in the process. In a vector notation, when force vector and displacement vector are given it is the dot product of both. Such integrals are called line integrals. 6.2.4 Describe the flux and circulation of a vector field. Work done is path independent if the force is a conservative force, meaning there is a potential scalar function (this is all in classical physics) whose negative gradient is the force. As taught during our Physics tuition classes, work done by a force is given as the product of the magnitude of the force and that of displacement. 5.1 Line integrals in two dimensions Instead of integrating over an interval a,b we can integrate over a curve C. 6.2.3 Use a line integral to compute the work done in moving an object along a curve in a vector field. The concept of line integral and its application in physics with a simple example is elucidated in the current article. ![]() We want to find the work done between positions A and B, so. ![]() We also have discussed an example to understand its application in the world of physics. Let’s discuss something more specific that comes under the larger concept of integral calculus. Now, to find the total work done, we add up all the little portions of d W, which is what take an integral is. Green’s theorem to write this line integral as a double integral with the. We have discussed in earlier articles what integral calculus is and how it works. path that the points (0,0), (2,2), and (0,2) in a counterclockwise manner. The role of integral calculus comes in areas ranging from classical mechanics to quantum mechanics. ![]() How can we use the formula giving a type I integral to compute the length of the curve $\gamma$? Simple, we just integrate a constant funciton $f(x, y) = 1$, this works exactly how we can compute the length of a segment by integrating $f(x)=1$ in a standard integral.Integral calculus and physics are inseparable. Let's now look at an example to discover one of the simplest and yet useful application of line integrals, namely computing the length of a curve or function. ![]()
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