Question

The potential energy for a force field $\overrightarrow{F}$ is given by $U(x,y)=\cos (x+y)$. The force acting on a particle at the position given by coordinates $(0,\frac{\pi }{4})$ is

• A. $-\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
• B. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
• C. $(\frac{1}{2}\hat{i}+\frac{\sqrt{3}}{2}\hat{j})$
• D. $(\frac{1}{2}\hat{i}-\frac{\sqrt{3}}{2}\hat{j})$

Answer 1

The correct option is B $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$

Given,
$U(x,y)=\cos (x+y)$
Now, the relation betweeen Conservative force and Potential energy is given by $\overrightarrow{F}=-(\frac{\partial U}{\partial x}\hat{i}+\frac{\partial U}{\partial y}\hat{j}+\frac{\partial U}{\partial z}\hat{k})$
Since, the given potential energy is a function of $x$ and $y$.
we can write that,
$\overrightarrow{F}=F_x\hat{i}+F_y\hat{j}=-(\frac{\partial U}{\partial x}\hat{i}+\frac{\partial U}{\partial y}\hat{j}).........\left(1\right)$
$X-$ component of force:
$F_x=-\frac{\partial U}{\partial x}=\sin (x+y)=\frac{1}{\sqrt{2}}$
at $(0,\frac{\pi }{4})$
$Y-$ component of force:
$F_y=-\frac{\partial U}{\partial y}=\sin (x+y)=\frac{1}{\sqrt{2}}$
at $(0,\frac{\pi }{4})$
$\overrightarrow{F}=\frac{1}{\sqrt{2}}[\hat{i}+\hat{j}]$
Thus, option (b) is the correct answer.