The potnetial energy for a force field `vecF` is given by `U(x, y)=cos(x+y)`. The force acting on a particle at the position given by coordinates `(0, pi//4)` is

Options

A. `-(1)/(sqrt2)(hati+hatj)`
B. `(1)/(sqrt2)(hati+hatj)`
C. `((1)/(2)hati+(sqrt3)/(2)hatj)`
D. `((1)/(2)hati-(sqrt3)/(2)hatj)`

Answer

To find the force acting on a particle at the position given by coordinates (0, π/4), we start with the potential energy function \( U(x, y) = \cos(x + y) \).

### Step 1: Determine the force components

The force \( \vec{F} \) in a conservative force field can be derived from the potential energy using the following relationships:

\[ \vec{F} = -\nabla U \]

This means that the components of the force can be calculated as:

\[ F_x = -\frac{\partial U}{\partial x} \]
\[ F_y = -\frac{\partial U}{\partial y} \]

### Step 2: Calculate \( F_x \)

First, we need to compute \( \frac{\partial U}{\partial x} \):

\[ U(x, y) = \cos(x + y) \]

Using the chain rule:

\[ \frac{\partial U}{\partial x} = -\sin(x + y) \]

Now, we evaluate this at the point \( (0, \frac{\pi}{4}) \):

\[ F_x = -\left(-\sin(0 + \frac{\pi}{4})\right) = \sin\left(\frac{\pi}{4}\right) \]

Since \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):

\[ F_x = \frac{1}{\sqrt{2}} \]

### Step 3: Calculate \( F_y \)

Next, we compute \( \frac{\partial U}{\partial y} \):

\[ \frac{\partial U}{\partial y} = -\sin(x + y) \]

Evaluating this at the same point \( (0, \frac{\pi}{4}) \):

\[ F_y = -\left(-\sin(0 + \frac{\pi}{4})\right) = \sin\left(\frac{\pi}{4}\right) \]

Again, since \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):

\[ F_y = \frac{1}{\sqrt{2}} \]

### Step 4: Combine the components to find the force vector

Now we can express the force vector \( \vec{F} \):

\[ \vec{F} = F_x \hat{i} + F_y \hat{j} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]

### Final Answer

Thus, the force acting on the particle at the position \( (0, \frac{\pi}{4}) \) is:

\[ \vec{F} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]

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