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Question: Which of the following forces is conservative? A. \(\overrightarrow F = y\widehat i - x\widehat j\...

Which of the following forces is conservative?
A. F=yi^xj^\overrightarrow F = y\widehat i - x\widehat j
B. F=xyi^xyj^\overrightarrow F = xy\widehat i - xy\widehat j
C. F=yi^+xj^\overrightarrow F = y\widehat i + x\widehat j
D. F=x2yi^y2xj^\overrightarrow F = {x^2}y\widehat i - {y^2}x\widehat j

Explanation

Solution

In order to find a conservative force we have to find the partial derivative with respect to x and y. Then we have to make a matrix, where the partial derivative will form a column, Force’s x and y part another column and dimension the other column. The determinant of the option which comes out to be 00 is the conservative force.

Formula used:
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ {\dfrac{{\partial F}}{{\partial x}}}&{\dfrac{{\partial F}}{{\partial y}}}&{\dfrac{{\partial F}}{{\partial z}}} \\\ {{F_x}}&{{F_y}}&{{F_z}} \end{array}} \right) is the matrix whose determinant is to be found. Fx\dfrac{{\partial F}}{{\partial x}} is the partial derivative of force F\overrightarrow F with respect to xx,Fy\dfrac{{\partial F}}{{\partial y}} is the partial derivative of force F\overrightarrow F with respect to yyand Fz\dfrac{{\partial F}}{{\partial z}} is the partial derivative of force F\overrightarrow F with respect to zz. Fx{F_x} is the force in xx-axis, Fy{F_y} is the force in yy- axis and Fz{F_z} is the force in zz- axis.

Complete step by step answer:
(A) F=yi^xj^\overrightarrow F = y\widehat i - x\widehat j we get,
Fx=y{F_x} = y and Fy=x{F_y} = - x,
Now the partial derivative is,
Fx=0\dfrac{{\partial F}}{{\partial x}} = 0 and Fy=0\dfrac{{\partial F}}{{\partial y}} = 0
Now, by putting them in matrix we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ {\dfrac{{\partial F}}{{\partial x}}}&{\dfrac{{\partial F}}{{\partial y}}}&{\dfrac{{\partial F}}{{\partial z}}} \\\ {{F_x}}&{{F_y}}&{{F_z}} \end{array}} \right)
Substituting the values we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ 0&0&0 \\\ y&{ - x}&0 \end{array}} \right)
The determinant of the given matrix is 00. Hence F=yi^xj^\overrightarrow F = y\widehat i - x\widehat j is a conservative force.

(B) F=xyi^xyj^\overrightarrow F = xy\widehat i - xy\widehat j,
Fx=xy{F_x} = xy and Fy=xy{F_y} = - xy,
Now the partial derivative is,
Fx=y\dfrac{{\partial F}}{{\partial x}} = y and Fy=x\dfrac{{\partial F}}{{\partial y}} = - x
Now, by putting them in matrix we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ {\dfrac{{\partial F}}{{\partial x}}}&{\dfrac{{\partial F}}{{\partial y}}}&{\dfrac{{\partial F}}{{\partial z}}} \\\ {{F_x}}&{{F_y}}&{{F_z}} \end{array}} \right)
Substituting the values we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ y&{ - x}&0 \\\ {xy}&{ - xy}&0 \end{array}} \right)
The determinant of the given matrix is (y2x+x2y)k^( - {y^2}x + {x^2}y)\widehat k. Hence F=xyi^xyj^\overrightarrow F = xy\widehat i - xy\widehat j is not a conservative force.

(C) F=yi^+xj^\overrightarrow F = y\widehat i + x\widehat j we get,
Fx=y{F_x} = y and Fy=x{F_y} = x,
Now the partial derivative is,
Fx=0\dfrac{{\partial F}}{{\partial x}} = 0 and Fy=0\dfrac{{\partial F}}{{\partial y}} = 0
Now, by putting them in matrix we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ {\dfrac{{\partial F}}{{\partial x}}}&{\dfrac{{\partial F}}{{\partial y}}}&{\dfrac{{\partial F}}{{\partial z}}} \\\ {{F_x}}&{{F_y}}&{{F_z}} \end{array}} \right)
Substituting the values we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ 0&0&0 \\\ y&x;&0 \end{array}} \right)
The determinant of the given matrix is 00. Hence F=yi^+xj^\overrightarrow F = y\widehat i + x\widehat j is a conservative force.

(D) F=x2yi^y2xj^\overrightarrow F = {x^2}y\widehat i - {y^2}x\widehat j we get,
Fx=x2y{F_x} = {x^2}y and Fy=xy2{F_y} = - x{y^2},
Now the partial derivative is,
Fx=2xy\dfrac{{\partial F}}{{\partial x}} = 2xy and Fy=2xy\dfrac{{\partial F}}{{\partial y}} = - 2xy
Now, by putting them in matrix we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ {\dfrac{{\partial F}}{{\partial x}}}&{\dfrac{{\partial F}}{{\partial y}}}&{\dfrac{{\partial F}}{{\partial z}}} \\\ {{F_x}}&{{F_y}}&{{F_z}} \end{array}} \right)
Substituting the values we get,
\left( {\begin{array}{*{20}{c}} {\widehat i}&{\widehat j}&{\widehat k} \\\ {2xy}&{ - 2xy}&0 \\\ {{x^2}y}&{ - x{y^2}}&0 \end{array}} \right)
The determinant of the given matrix is (2x2y3+2x3y2)k^( - 2{x^2}{y^3} + 2{x^3}{y^2})\widehat k. Hence F=x2yi^y2xj^\overrightarrow F = {x^2}y\widehat i - {y^2}x\widehat j is not a conservative force.

Note: Alternative method: It is for only two-dimensional vector if dFxdy=dFydx\dfrac{{d{F_x}}}{{dy}} = \dfrac{{d{F_y}}}{{dx}} then it is also conservative, where is the force in xx-axis and Fy{F_y} in yy-axis. It is necessary to partially derive the terms. It is very important to separate the whole Force into the terms of xx,yy and zz. Till the determinant of any vector is not found out to be 00 then it is not conservative.