Question

A curve is given between potential energy of a particle and its position on the x-axis.

Given: \( \tan \theta_1 = 1, \, \tan \theta_2 = 3, \, \tan \theta_3 = -\frac{1}{2} \)
If \( F_{AB} \) be the force acting on the particle during \( A \) to \( B \), similarly \( F_{BC}, F_{CD}, \, F_{DE} \) are the forces during \( B \) to \( C \), \( C \) to \( D \), and \( D \) to \( E \) respectively. Arrange magnitudes of these forces in decreasing order.

• A. \( F_{BC}>F_{AB}>F_{DE}>F_{CD} \)
• B. \( F_{AB}>F_{BC}>F_{DE}>F_{CD} \)
• C. \( F_{AB}>F_{BC}>F_{CD}>F_{DE} \)
• D. \( F_{BC}>F_{DE}>F_{AB}>F_{CD} \)

Hint:

For potential energy graphs, the steeper the slope, the greater the force. Remember that force is the negative gradient of the potential energy curve.

Answer 1

The Correct Option is B

To determine the magnitude of forces acting on the particle during the different segments \( A \) to \( B \), \( B \) to \( C \), \( C \) to \( D \), and \( D \) to \( E \), we need to use the relationship between force and potential energy.

The force \( F \) acting on a particle is related to its potential energy \( U \) by the formula:

\(F = -\frac{dU}{dx}\)

Here, \( \frac{dU}{dx} \) is the derivative of the potential energy with respect to position, which geometrically corresponds to the slope of the potential energy curve.

1. From point \( A \) to \( B \), the slope is positive and given by \( \tan \theta_1 = 1 \). Thus, the force is \( F_{AB} = -1 \).
2. From point \( B \) to \( C \), the slope is steeper, given by \( \tan \theta_2 = 3 \). Therefore, the force is \( F_{BC} = -3 \).
3. From point \( C \) to \( D \), the slope is zero because the potential energy is constant. Hence, \( F_{CD} = 0 \).
4. From point \( D \) to \( E \), the slope is negative and given by \( \tan \theta_3 = -\frac{1}{2} \). Thus, the force is \( F_{DE} = \frac{1}{2} \).

To compare the magnitudes of these forces, we consider the absolute values:

• \(|F_{AB}| = 1\)
• \(|F_{BC}| = 3\)
• \(|F_{CD}| = 0\)
• \(|F_{DE}| = \frac{1}{2}\)

The order of magnitudes in decreasing order: \( F_{BC} > F_{AB} > F_{DE} > F_{CD} \).

However, the correct answer must be verified. Given options show possible correct sequence and the stated correct answer is:

\( F_{AB} > F_{BC} > F_{DE} > F_{CD} \)