Question
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
B. F=xyi−xyj
C. F=yi+xj
D. F=x2yi−y2xj
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 0 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. ∂x∂F is the partial derivative of force F with respect to x,∂y∂F is the partial derivative of force F with respect to yand ∂z∂F is the partial derivative of force F with respect to z. Fx is the force in x-axis, Fy is the force in y- axis and Fz is the force in z- axis.
Complete step by step answer:
(A) F=yi−xj we get,
Fx=y and Fy=−x,
Now the partial derivative is,
∂x∂F=0 and ∂y∂F=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 0. Hence F=yi−xj is a conservative force.
(B) F=xyi−xyj,
Fx=xy and Fy=−xy,
Now the partial derivative is,
∂x∂F=y and ∂y∂F=−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. Hence F=xyi−xyj is not a conservative force.
(C) F=yi+xj we get,
Fx=y and Fy=x,
Now the partial derivative is,
∂x∂F=0 and ∂y∂F=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 0. Hence F=yi+xj is a conservative force.
(D) F=x2yi−y2xj we get,
Fx=x2y and Fy=−xy2,
Now the partial derivative is,
∂x∂F=2xy and ∂y∂F=−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. Hence F=x2yi−y2xj is not a conservative force.
Note: Alternative method: It is for only two-dimensional vector if dydFx=dxdFy then it is also conservative, where is the force in x-axis and Fy in y-axis. It is necessary to partially derive the terms. It is very important to separate the whole Force into the terms of x,y and z. Till the determinant of any vector is not found out to be 0 then it is not conservative.