Understanding Potential Energy Function through Graph Analysis

[sonutulsiani]

Homework Statement

You are given the plot of a potential energy function U(x) versus x.

http://bcs.whfreeman.com/WebPub/Physics/tiplerphysics6e_bridge/question_bank_images/7-8.png?3200

1. For each point indicated, state whether the x component of the force associated with this function is positive (+), negative(−), or zero (0)), by matching the points with the corresponding sign.
1. Zero
2. Negative
3. Positive

A = ?
B = ?
C = ?
D = ?
E = ?
F = ?



2. At which point does the force have the greatest magnitude?


3. Which point is a stable equilibrium point?

Homework Equations

The Attempt at a Solution

I don't know how to start. Please help.

 

[Gear300]

For conservative fields, which is usually what we take potential energy for, F = -dU/dx

 

[sonutulsiani]

I didn't understand what you said.

 

[Gear300]

The force is negative the rate of change of potential energy with respect to displacement. So you can evaluate it as negative the derivative of U with respect to x.

 

[sonutulsiani]

You mean we have to convert this graph into it's derivative? And then what?

 

[Gear300]

Well...you don't really have to convert it to its derivative...you can just imagine the tangents at the different points. Since force is negative the derivative, if the tangent has a negative slope, then the force is positive, and so on. The case of stable equilibrium is essentially when you have a restoring force - meaning that if you move to the left (- displacement), then the force is to the right (+ direction) and vice versa.

 

[sonutulsiani]

So

A=2
B=1
C=3
D=1
E=2
F=1

2nd question, force having greatest magnitude is point C ?

3rd question, is it point F ??

 

[ideasrule]

So

A=2
B=1
C=3
D=1
E=2
F=1

2nd question, force having greatest magnitude is point C ?

3rd question, is it point F ??

All of those answers are correct except the last one. Why F? For a stable equilibrium, you need to find a point where decreasing x causes a positive force, so U has to have a negative slope on the let. Increasing x must cause a negative force, so U has to have positive slope on the right. There's only one point where this is true.

 

[sonutulsiani]

Is it C then? Because it has a positive slope on the right and negative on the left.

 

[sonutulsiani]

C or E

 

[sonutulsiani]

It is E! Isn't it?

 

[ideasrule]

Is it C then? Because it has a positive slope on the right and negative on the left.

Actually, the slope is negative on both the left and the right. For E, the slope is positive on both left and right.

Another way to think about this: if you need to add potential energy to move away from a certain x value, that x value is a stable equilibrium point.

 

[sonutulsiani]

It's D

 

[sonutulsiani]

is it?

 

[Gear300]

Yup.

 

[sonutulsiani]

Ohh! Okk thank you so much! Thanks a lot! :)