Question

A time independent conservative force \( \vec{F} \) has the form, \( \vec{F} = 3y\hat{i} + f(x, y)\hat{j} \). Its magnitude at \( x = y = 0 \) is 8. The allowed form(s) of \( f(x, y) \) is (are):

• A. \( 3x + 8 \)
• B. \( 2x + 8(y - 1)^2 \)
• C. \( 3x + 8e^{-y^2} \)
• D. \( 2x + 8\cos y \)

Hint:

For conservative forces, ensure \( \frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x} \). Always integrate carefully and apply given boundary or magnitude conditions.

Answer 1

The Correct Option is A, C

Step 1: Apply the condition for a conservative force. 
For a conservative force \( \vec{F} = (F_x, F_y) = (3y, f(x, y)) \), the condition for conservativeness is \[ \frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}. \] Step 2: Compute partial derivatives. 
\[ \frac{\partial F_x}{\partial y} = 3, \quad \frac{\partial F_y}{\partial x} = \frac{\partial f}{\partial x}. \] Thus, \[ \frac{\partial f}{\partial x} = 3. \] Step 3: Integrate with respect to \( x \). 
\[ f(x, y) = 3x + g(y), \] where \( g(y) \) is an arbitrary function of \( y \). 
Step 4: Use the magnitude condition. 
At \( x = y = 0 \), the magnitude of the force is \[ |\vec{F}| = \sqrt{(3y)^2 + f^2} = |f(0, 0)| = 8. \] \[ f(0, 0) = g(0) = 8. \] Thus, \( g(y) \) must satisfy \( g(0) = 8. \) 
|Step 5: Possible form of \( f(x, y) \). 
The simplest allowed form is \( f(x, y) = 3x + 8 \). 
Step 6: Final Answer. 
Hence, the correct answer is \( f(x, y) = 3x + 8. \)