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Chapter 4: Problem 70

Potential energy associated with a conservative force is given by \(U=A x^{2}\), where \(A\) is a constant then (A) force always tends to accelerate the particle towards origin. (B) force always tends to accelerate the particle away from origin. (C) force always tends to accelerate the particle towards the origin if \(A\) is positive. (D) force always tends to accelerate the particle away from origin if \(A\) is negative.

Short Answer

Expert verified
The correct statement is: force always tends to accelerate the particle towards the origin if \(A\) is positive (option C).

Step by step solution

01

To determine the force from the potential energy function, we need to take the negative derivative of the given potential energy with respect to x: F(x) = - dU/dx #Step 2: Evaluate the derivative#

Now, we can evaluate the derivative of U = Ax^2 with respect to x: F(x) = - d (Ax^2) / dx Using the power rule , we obtain: F(x) = - 2Ax #Step 3: Determine the nature of force for various 'A' values#
02

From F(x)=-2Ax, we can analyze the behavior of the force by looking at the signs for positive and negative values of A. If A is positive: F(x) = -2Ax < 0, for any x > 0, and F(x) = -2Ax > 0, for any x < 0 If A is negative: F(x) = -2Ax > 0, for any x > 0, and F(x) = -2Ax < 0, for any x < 0 #Step 4: Identify the correct statement#

Based on the above results: (A) is incorrect, because force can also decelerate the particle towards the origin for A < 0 and x < 0. (B) is incorrect, because force can also decelerate the particle away from the origin for A < 0 and x > 0. (C) is correct, as force always tends to accelerate the particle towards the origin if A is positive (F(x) is opposite in sign to x). (D) is incorrect, because force can decelerate the particle away from the origin for A < 0 and x < 0. Thus, option (C) is the correct statement.

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