Question

The alternative that gives the conservative force of the following is:

• A. $\overrightarrow{F_1}=2xy\hat{i}+x^2\hat{j}$
• B. $\overrightarrow{F_2}=y^3\hat{i}+xy^2\hat{j}$
• C. $\overrightarrow{F_3}=y\hat{i}+x\hat{j}$
• D. $\overrightarrow{F_4}=xy^2\hat{i}+x^2\hat{j}$

Answer 1

Answer: (A), (C)

A and C

Answer 2

A force $\overrightarrow{F} = F_x \hat{i} + F_y \hat{j}$ is conservative if its curl is zero. For a two-dimensional force, this condition simplifies to $\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}$.

We check this condition for each option: (A) $\overrightarrow{F_1} = 2xy\hat{i} + x^2\hat{j}$: $F_x = 2xy$, $F_y = x^2$. $\frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y}(2xy) = 2x$. $\frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x}(x^2) = 2x$. Since $\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}$ (both are $2x$), $\overrightarrow{F_1}$ is a conservative force.

(B) $\overrightarrow{F_2} = y^3\hat{i} + xy^2\hat{j}$: $F_x = y^3$, $F_y = xy^2$. $\frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y}(y^3) = 3y^2$. $\frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x}(xy^2) = y^2$. Since $\frac{\partial F_y}{\partial x} \neq \frac{\partial F_x}{\partial y}$ ($y^2 \neq 3y^2$), $\overrightarrow{F_2}$ is not a conservative force.

(C) $\overrightarrow{F_3} = y\hat{i} + x\hat{j}$: $F_x = y$, $F_y = x$. $\frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y}(y) = 1$. $\frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x}(x) = 1$. Since $\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}$ (both are $1$), $\overrightarrow{F_3}$ is a conservative force.

(D) $\overrightarrow{F_4} = xy^2\hat{i} + x^2\hat{j}$: $F_x = xy^2$, $F_y = x^2$. $\frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y}(xy^2) = 2xy$. $\frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x}(x^2) = 2x$. Since $\frac{\partial F_y}{\partial x} \neq \frac{\partial F_x}{\partial y}$ ($2x \neq 2xy$), $\overrightarrow{F_4}$ is not a conservative force.

Therefore, the conservative forces are $\overrightarrow{F_1}$ and $\overrightarrow{F_3}$.