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Chapter 4: Problem 437

The potential energy of a body is given by \(\mathrm{U}=\mathrm{A}-\mathrm{Bx}^{2}\) (where \(\mathrm{x}\) is displacement). The magnitude of force acting on the particle is (A) constant (B) proportional to \(\mathrm{x}\) (C) proportional to \(\mathrm{x}^{2}\) (D) Inversely proportional to \(\mathrm{x}\)

Short Answer

Expert verified
The magnitude of the force acting on the particle is proportional to x (Option B), as the force acting on the particle is given by \(F = 2Bx\), where F is the force, B is a constant, and x is the displacement.

Step by step solution

01

Write down the potential energy function

The given potential energy function is \(U = A - Bx^2\), where U is the potential energy, A and B are constants, and x is the displacement.
02

Differentiate the potential energy function with respect to x

We need to find the negative gradient of the potential energy function with respect to x. This is the same as finding the first derivative of the given function with respect to x and multiplying it by -1. Differentiate the function with respect to x: \( \frac{dU}{dx} = \frac{d(A - Bx^2)}{dx} \) Using the power rule, we get: \( \frac{dU}{dx} = -2Bx \) Now, multiply by -1: \( F = -\frac{dU}{dx} = 2Bx \)
03

Analyze the relationship between force and displacement

Now, we have the force acting on the particle: \(F = 2Bx\) Comparing this with the given options, we can eliminate options (A) and (C) since the force is neither constant nor proportional to x^2. Since the force is directly proportional to x, we can eliminate option (D) as well. Therefore, the correct answer is: (B) proportional to x

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