First slide

Concept of potential energy

Question

The potential energy of a force field is given by $U = \sin (x + y)$ . The force acting on the particle at position $(\frac{π}{4}, \frac{π}{4})$ is

Moderate

A

$- \frac{1}{\sqrt{2}} (\hat{i} + \hat{j})$
B

$\frac{1}{\sqrt{2}} (\hat{i} + \hat{j})$
C

$- \frac{1}{\sqrt{2}} \hat{i}$
D

0

Solution

$\begin{array}{l} \vec{F} = \frac{- \partial U}{\partial x} \hat{i} - \frac{\partial U}{\partial y} \hat{j} - \frac{\partial U}{\partial z} \hat{k} \\ \vec{F} = - \cos (x + y) \hat{i} - \cos (x + y) \hat{j} \\ \vec{F} = - \cos (x + y) (\hat{i} + \hat{j}) \\ At (\frac{π}{4}, \frac{π}{4}) \\ \vec{F} = - \cos (\frac{π}{4} + \frac{π}{4}) \vec{(i} + \hat{j}) \\ \vec{F} = \vec{0} F = 0 \end{array}$

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