Work done by gravity on a hanging chain?

[Manish_529]

Homework Statement

A chain of length 'L' and mass 'M' and also has an uniform linear mass density. The chain hangs from a support as one of it's end is attached to it, while the other end is free and is hanging downwards. Find the work done by gravity on the entire hanging chain.

Relevant Equations

W=F.S

I tried taking an element of length dx and tried calculating the force of gravity acting on it so that I could just integrate over the whole chain, but I couldn't figure out what to do of that displacement part since the dx element is not moving as is just at rest (hanging). So, how should I proceed with the integration? (A picture has been attached for reference of the above situation.)

 

[phinds]

If the chain is just hanging there, is any work being done?

 

[Manish_529]

since, there is a difference in the P.E of the 2 ends of the chain, there should be a work done cause the work done by conservative forces (which here is just gravity)= -(change in P.E)

 

[jbriggs444]

since, there is a difference in the P.E of the 2 ends of the chain, there should be a work done cause the work done by conservative forces (which here is just gravity)= -(change in P.E)

Work is done when there is a change in potential energy over time.

A change in potential energy over displacement does not qualify as work and would not have the same units as work.

In an advanced class, you may learn that change in a potential over displacement is called the "gradient" of the potential. For a scalar potential (like gravity) the gradient is a vector. In the case of gravitational potential the gradient is the local acceleration of gravity.

It would be meaningful to ask how much work was done by a chain that was laying motionless on a horizontal frictionless surface and then slowly lowered through a frictionless hole in the ceiling to end up in the pictured configuration.

 

[phinds]

since, there is a difference in the P.E of the 2 ends of the chain, there should be a work done cause the work done by conservative forces (which here is just gravity)= -(change in P.E)

But if it is just hanging there, there IS no change in P.E. --- as jbriggs said, you need a change in P.E. OVER TIME.

 

[Manish_529]

Work is done when there is a change in potential energy over time.

A change in potential energy over displacement does not qualify as work and would not have the same units as work.

In an advanced class, you may learn that change in a potential over displacement is called the "gradient" of the potential. For a scalar potential (like gravity) the gradient is a vector. In the case of gravitational potential the gradient is the local acceleration of gravity.

It would be meaningful to ask how much work was done by a chain that was laying motionless on a horizontal frictionless surface and then slowly lowered through a frictionless hole in the ceiling to end up in the pictured configuration.

Work is done when there is a change in potential energy over time.

A change in potential energy over displacement does not qualify as work and would not have the same units as work.

In an advanced class, you may learn that change in a potential over displacement is called the "gradient" of the potential. For a scalar potential (like gravity) the gradient is a vector. In the case of gravitational potential the gradient is the local acceleration of gravity.

It would be meaningful to ask how much work was done by a chain that was laying motionless on a horizontal frictionless surface and then slowly lowered through a frictionless hole in the ceiling to end up in the pictured configuration.

In case of stationary field like that of gravity the force at a point doesn't depend on the time factor rather it depends on the distance from the body applying it so the force is a function of position then the associated potential energy also must be a function of position meaning the potential energy associated with gravity changes with position and not time.

 

[phinds]

changes with position and not time.

That's quite a trick, changing something's position instantaneously (i.e. taking no time)

 

[Manish_529]

Oh i get it work is done by a force when a body change's it's position with the passage of time which in turn changes the P.E with the position because the body's position was changed. Am I right?

 

[jbriggs444]

Oh i get it work is done by a force when a body change's it's position with the passage of time which in turn changes the P.E with the position because the body's position was changed. Am I right?

Yes. This will do.

Often, the idea of potential energy will be introduced with point-like objects so that this confusion cannot arise. A point-like object cannot be in two places at once. So if there is a change of potential energy, there must be a situation before and a situation after. Some time must have elapsed.

If we allow the possibility of extended objects then we can compute the potential energy for the object by adding up the potential energies for all of the object's parts. We may have to evaluate an integral.

If we want to apply the work energy theorem and equate the change in potential energy with the amount of mechanical work that is performed then we might have to evaluate two integrals and subtract them from each other do determine the change in potential energy.

 

[Steve4Physics]

Oh i get it work is done by a force when a body change's it's position with the passage of time which in turn changes the P.E with the position because the body's position was changed. Am I right?

More or less right but can I add this...

Suppose you have an object weighing 20N. That means the object experiences a gravitational force (its weight) of 20N acting downwards.

If the object moves down by 3m (in the same direction to the force’s direction) the work done by the weight = 20N x 3m = 60J. The change in gravitational potential energy (GPE) is the negative of this, i.e. -60J. So GPE reduces by 60J

If the object moves up by 3m (in the opposite direction to the force’s direction) the work done by the weight = - 20N x 3m = -60J. The change in gravitational potential energy (GPE) is the negative of this, i.e. 60J. So GPE increases by 60J.

But if the object moves horizontally sideways by 3m (perpendicular to the force’s direction, e.g. slides along a horizontal table), the work done by the weight = zero because there is no motion in/against the direction of the weight. The change in GPE is zero.

If the displacement and weight are at some arbitrary angle (e.g. a box sliding along a tilted ramp) a bit of trigonometry is needed.

 

[Manish_529]

but then how is the potential energy of the chain varying if there is no work being done by the gravity?

 

[haruspex]

but then how is the potential energy of the chain varying if there is no work being done by the gravity?

I'm not sure which statement you are responding to there. Please use the Reply or Quote feature so readers know what you are asking about.

Let's start with the simple case of an object in free fall. Gravity does work on the object. Gravity obtains the work from the GPE, so that reduces. The work done on the object takes the form of KE, so that increases.
If, instead, the object is lowered at constant speed, some other force, equal to the weight of the object, is supporting it and doing negative work on the object equal in magnitude to the work done by gravity: equal and opposite force, same displacement.

 

[jbriggs444]

but then how is the potential energy of the chain varying if there is no work being done by the gravity?

Are you asking how the potential energy of a 10 cm length of chain at the top, near the ceiling can be different from the potential energy of a 10 cm length of chain at the bottom, farther from the ceiling?

Answer: They are different because the two lengths are at different average heights.

Are you asking why no work is done by gravity?

Answer: Because there is no motion. The chain is hanging motionless.

You could ask how much work would be performed by gravity if the 10 cm length of chain near the top was slowly lowered to the same height as the 10 cm length of chain at the bottom.

Answer: The work done by gravity on that 10 cm length of chain would be equal to the difference in the two potential energies.

 

[Manish_529]

It's actually the first one how can two points have different potential energies in a force field if no work is being done by that force

 

[jbriggs444]

It's actually the first one how can two points have different potential energies in a force field if no work is being done by that force

What definition of potential energy are you using?

I am trying to understand why you think that work must be continuously supplied by a force in order for potential energy differences to exist. The way to do that is to go back to definitions and first principles and unravel your mistaken understanding.

 

[haruspex]

It's actually the first one how can two points have different potential energies in a force field if no work is being done by that force

If the mass at each point is static then no work is being done on either. 0+0=0.

 

[Manish_529]

What definition of potential energy are you using?

I am trying to understand why you think that work must be continuously supplied by a force in order for potential energy differences to exist. The way to do that is to go back to definitions and first principles and unravel your mistaken understanding.

The difference in potential energy of a body is equal to the work done by conservative forces on it

 

[jbriggs444]

The difference in potential energy of a body is equal to the work done by conservative forces on it

Not exactly.

The difference in potential energy between two states for a single body is equal to the work that would have been done by an external force to move it slowly from the initial state to the final state.

So for instance, the difference between the potential energy of a sled at the top of a hill and the same sled at the bottom of the hill is equal to the work that a child would need to do on the sled, dragging it slowly up the hill over frictionless snow.

In the case of the hanging chain, you have not identified an initial state and a final state.

 

[Steve4Physics]

It's actually the first one how can two points have different potential energies in a force field if no work is being done by that force

Points don't have 'potential energies', they have ‘potentials’. It’s important to distinguish between ‘potential’ ($V$) and ‘potential energy’ ($E$). Maybe that's part of the difficulty.

E.g.
Point A has a gravitational potential $V_A = 2 J/kg$.
Point B has a gravitational potential $V_B = 5 J/kg$.
If a 10 kg mass is moved from A to B, the change in gravitational potential energy is:
$\Delta E = m\Delta V = 10 kg \times (5 J/kg - 2 J/kg) = 30 J$
____________

We can choose the position where $V=0$ for convenience. At an introductory level, we often take ground-level as $V=0$. In this case, the potential of a point at height $h$ above the ground is $gh$ so the potential energy of a mass at height $h$ is $mgh$.

Now go back to your orighinal question...

E.g.
A 5 kg chain is initially laying flat on the ground.The chain is then hung from the ceiling with the chain’s centre of gravity 2m above the ground.

The change in the chain’s gravitational potential energy is:
$\Delta E = m \Delta V = m(gh – 0) = mgh = 5 kg \times 10 N/kg \times 2m = 100 J$

 

[Manish_529]

Not exactly.

The difference in potential energy between two states for a single body is equal to the work that would have been done by an external force to move it slowly from the initial state to the final state.

So for instance, the difference between the potential energy of a sled at the top of a hill and the same sled at the bottom of the hill is equal to the work that a child would need to do on the sled, dragging it slowly up the hill over frictionless snow.

In the case of the hanging chain, you have not identified an initial state and a final state.

so, for a hanging chain what will be it's potential energy, and how do we define potential energies for such scenarios

 

[phinds]

so, for a hanging chain what will be it's potential energy, and how do we define potential energies for such scenarios

You have to define a base. If the bottom of the chain is 10 feet off the ground (we normally use that as a base) and the chain is 2 feet long, then the potential energy is based on 11 feet off the ground (center of gravity) and the weight of the chain.

The bottom of the chain (and likewise the top) is a point and since a point has no weight then, as @Steve4Physics already explained to you, there IS NO "potential energy", just potential.

 

[Steve4Physics]

so, for a hanging chain what will be it's potential energy, and how do we define potential energies for such scenarios

Already explained in Post #19.

 

[jbriggs444]

You have to define a base.

This is an important point about potentials. And about potential energy. Only potential energy differences are physicically meaningful.

The "potential energy" of an object is the work that would need to be done on the object to move it from its base location (sometimes called a reference location) to its actual location.

We are free to adopt a reference location anywhere we please. Though some choices are so obvious and convenient that they become standard.

One could adopt a reference level at the ceiling. A chain hanging below the ceiling would have potential energy given by $mgh$ where $m$ is the mass of the chain, $g$ is the local acceleration of gravity and $h$ is the displacement from the ceiling to the center of gravity. In this case, $h$ would be negative and so would be the potential energy of the chain.

One could adopt a reference level at the floor. A chain hanging below the ceiling would have potential energy given by $mgh$ where $m$ is the mass of the chain, $g$ is the local acceleration of gravity and $h$ is the displacement from the floor to the center of gravity. In this case, $h$ would be positive and so would be the potential energy of the chain.

One could adopt a reference level at the street outside. If the room is in the basement, the potential energy of the chain might be negative. If it is on the tenth floor, its potential energy would be positive.