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

Step 1: Force is the negative derivative of potential energy.
The force acting on the particle is given by \( F = - \frac{dU}{dx} \).
Step 2: Understanding the graph.
From the given graph, the slope of the potential energy curve gives the magnitude of the force. The steeper the slope, the greater the force.
Step 3: Compare slopes.
The slope of the potential energy curve is greatest between \( A \) to \( B \), followed by \( B \) to \( C \), then \( D \) to \( E \), and the least between \( C \) to \( D \). Therefore, the order of forces is: \[ F_{AB}>F_{BC}>F_{DE}>F_{CD} \] Step 4: Conclusion.
The correct answer is option (2).