Is the Force Conservative in a One-Dimensional System?

[atlantic]

Homework Statement

An atom is moving in a one-dimensional system (x on the horisontal axis):

A force, F is acting in the range -2m< x < 2m and can be written as:

F = -Cv​

where C is a constant, and v is the velocity of the atom. Is the force, F conservative?

Homework Equations

The force is conservative if it only depends on the end points of the motion.

The Attempt at a Solution

The force, F is a function of the velocity. Does that explicitly mean the the force is not conservative?

 

[ashishsinghal]

According to my calculations, when we convert v into x, we get that F is directly proportional to x. Hence it should be conservative.

 

[atlantic]

According to my calculations, when we convert v into x, we get that F is directly proportional to x. Hence it should be conservative.

Can you explain how you "convert v to x"?

 

[ehild]

According to my calculations, when we convert v into x, we get that F is directly proportional to x.

Why do you think so? The velocity -coordinate function includes an integration constant. At start, the velocity is arbitrary and so is F, being proportional to the velocity.

Velocity-dependent forces are not conservative.ehild

 

[atlantic]

Why do you think so? The velocity -coordinate function includes an integration constant. At start, the velocity is arbitrary and so is F, being proportional to the velocity.

Velocity-dependent forces are not conservative.

ehild

So I should conclude that the force is not conservative?

 

[ehild]

It depends what is required. Maybe you have to show that the work of the force between the same two points depends on the way the particle moves. Find the relation between F and x and calculate the work W = integral (Fdx). If this expression includes some other parameter(s) than the initial and final positions and the constant c, the work can not be obtained as difference of potential energies.

ehild

 

[ashishsinghal]

F = -cv
dv/dt = -cv/m
dv/dt*dx/dx = -cv/m
dv/dx*v = -cv/m
dv/dx = -c/m

v= -cx/m + k

now putting v = -cx/m + k
we get
F = c^2.x/m - ck
Sorry, F is not proportional to x but it depends on x.
Is there anything wrong in it?

 

[ehild]

Your derivation is correct. But you see that because of the parameter k, the force is not a well-defined, unique function of x. k is the velocity at x=0, and it is arbitrary. The work done between two points depends on the initial velocity, so it can not be obtained as the difference of potential energies belonging to those points.

ehild

 

[ashishsinghal]

Sorry, but I can't get you. Even though k is arbitrary. But k is a constant. Hence each value of k would be a new case and every such case would have conservative force.

 

[ehild]

In case of a conservative force, it does not matter how a body moves from A to B. If a body moves downward with h meters, no matter if it fall or you kept it move with constant velocity, the work of gravity is the same. The work is the same if the stone fall or slide down on an incline or carried down by a person.

In this problem, you got the force in terms of x, but it is valid only when there are no other forces acting on the body. Even then, both the force and work depend on the initial velocity.

If there are other forces acting on the body, for example one of equal magnitude and of opposite sign which keeps it moving with constant velocity then F=-Cv is constant. You can make F almost zero or as great as you like and the work of F, too.

ehild

 

[ashishsinghal]

If there are other forces acting on the body, for example one of equal magnitude and of opposite sign which keeps it moving with constant velocity then F=-Cv is constant. You can make F almost zero or as great as you like and the work of F, too.

ehild

But here the question is that if F is conservative. So you need to consider a case in which other forces do not act. If there is drag acting on a ball while free falling, it does not make gravity unconservative

 

[ehild]

As you said, an other force applied does not make the force non-conservative. Apply an equal and opposite force to F=-Cv, so the body moves 1 m distance with 1m/s positive velocity. The work of F is C. Now I make the body move 1 m again, between the same points as before, with 2 m/s velocity. The work of F is 2C. If it is a conservative force the work of F should be the same.
F=-Cv is a typical drag force. It is non-conservative.
Friction along a plane is a constant function in terms of the position. Is friction a conservative force? ehild

 

[ashishsinghal]

I got you, thanks