Elastic and gravitational potential energy

[Doc Al]

It reduces linearly, so the plot of force versus distance would be a dropping line, the energy would be the area under this which is 0.5F*x.

Good. (Where F is the weight and x the full stretch.) Is that work positive or negative? And how does it compare to the work done by gravity?

 

[Pratik89]

Good. (Where F is the weight and x the full stretch.) Is that work positive or negative? And how does it compare to the work done by gravity?

it should be positive because the direction of force and motion is the same and I think it would be half of the work done by gravity

 

[haruspex]

it should be positive because the direction of force and motion is the same and I think it would be half of the work done by gravity

Right, but how did you decide it was half?

 

[lychette]

My question is why is it exactly half and where does the other half end up ?

It is only half if force is proportional to extension (straight line graph of F against x )

 

[Doc Al]

it should be positive because the direction of force and motion is the same

What's the direction of the force exerted by the hand? What's the direction of motion?

 

[Pratik89]

What's the direction of the force exerted by the hand? What's the direction of motion?

the hand is not exerting a force, the hand is countering the force because of the force exerted by the mass, the hand initially exerts a force (because we are assuming that the body is always in equilibrium) and this is countered by the force exerted by the mass. As the mass drops, it requires progressively lesser support from the hand and finally when the body has dropped, the support required from the hand is zero. At every point the force exerted by the mass is balanced by supporting force from the hand. (while the motion takes place, the force because of the mass is infinitesimally higher than the supporting force and hence the downward motion)

 

[Doc Al]

the hand is not exerting a force

Of course it does. You describe that force yourself!

the hand is countering the force because of the force exerted by the mass, the hand initially exerts a force (because we are assuming that the body is always in equilibrium) and this is countered by the force exerted by the mass. As the mass drops, it requires progressively lesser support from the hand and finally when the body has dropped, the support required from the hand is zero. At every point the force exerted by the mass is balanced by supporting force from the hand. (while the motion takes place, the force because of the mass is infinitesimally higher than the supporting force and hence the downward motion)

There are three forces acting on the mass as it lowers. The upward force of the hand is one of those forces.

 

[Pratik89]

The mass is acted upon by these forces
1. It's own weight force
2. At the beginning the rope does not apply any force, and the hand completely supports it. As the mass is lowered, the support required from the hand reduces because the rope takes over. Finally, when the mass is at the lowest point the hand provides no support and all the force is from the rope.
I think we are going off track, this still does not answer the difference between the change in gravitational potential energy and storage of elastic energy in the rope.

 

[Doc Al]

I think we are going off track, this still does not answer the difference between the change in gravitational potential energy and storage of elastic energy in the rope.

Only because, for some reason, you are ignoring the work done by the hand.

 

[Doc Al]

The mass is acted upon by these forces
1. It's own weight force
2. At the beginning the rope does not apply any force, and the hand completely supports it. As the mass is lowered, the support required from the hand reduces because the rope takes over. Finally, when the mass is at the lowest point the hand provides no support and all the force is from the rope.

This is confusing.

There are three forces acting on the object as it is lowered:
1) the object's weight (a constant)
2) the upward force of the spring (which varies from 0 to max)
3) the upward force of the hand (which varies from max to 0)

 

[Pratik89]

This is confusing.

There are three forces acting on the object as it is lowered:
1) the object's weight (a constant)
2) the upward force of the spring (which varies from 0 to max)
3) the upward force of the hand (which varies from max to 0)

this is exactly what I mention in the second point, the zero to max variation ensures that that sum of forces in point 2 and point 3 adds up to the force in point 1.

 

[Doc Al]

this is exactly what I mention in the second point, the zero to max variation ensures that that sum of forces in point 2 and point 3 adds up to the force in point 1.

Sure. So?

Once again: Compare the work done by the hand with the work done by gravity. Then you'll see where the missing energy goes.

 

[haruspex]

this is exactly what I mention in the second point, the zero to max variation ensures that that sum of forces in point 2 and point 3 adds up to the force in point 1.

But you keep dodging the question... what is the work done by the hand in consequence of this? What is the displacement of the hand while exerting this force?

 

[Pratik89]

Sure. So?

Once again: Compare the work done by the hand with the work done by gravity. Then you'll see where the missing energy goes.

if your claim is that the energy goes to the hand, I am not entirely convinced. Work is defined as force*displacement. The net force on the hand is always zero. So no work has been done on the hand no matter how much it gets displaced

 

[haruspex]

The net force on the hand is always zero.

The force the mass exerts on the hand is not zero, therefore it does work on the hand. If the net force on the hand is zero it is because the hand is exerting a force on the arm, and so forth. The buck stops somewhere.

 

[Doc Al]

The net force on the hand is always zero. So no work has been done on the hand no matter how much it gets displaced

Let's talk about the work done on the object by the hand. (Work is done on the hand by the object. It exerts a force through a distance. Sure, it's not the only force acting on the hand. But so what?)

 

[Pratik89]

Let's talk about the work done on the object by the hand. (Work is done on the hand by the object. It exerts a force through a distance. Sure, it's not the only force acting on the hand. But so what?)

Alright, so this is what you are saying, correct me if I am wrong
The mass loses potential energy because it gets lower.
This potential energy is distributed to the rope and to the hand equally, because the hand is being acted upon by the mass (weight) of the object

 

[Doc Al]

Alright, so this is what you are saying, correct me if I am wrong
The mass loses potential energy because it gets lower.
This potential energy is distributed to the rope and to the hand equally, because the hand is being acted upon by the mass (weight) of the object

Careful with your terminology. But yes, the hand and the mass exert forces on each other, thus a portion of the energy goes into the hand (eventually becoming thermal energy). As you can work out, the gravitational energy divides equally between spring PE and work done on the hand.

 

[Pratik89]

Careful with your terminology. But yes, the hand and the mass exert forces on each other, thus a portion of the energy goes into the hand (eventually becoming thermal energy). As you can work out, the gravitational energy divides equally between spring PE and work done on the hand.

Alright, I have one alternative explanation too, but I am not sure it is correct
If the rope extends by x, the centre of mass of the rope should therefore move by x/2. Assuming that the center of mass (of rope) is the point where the new mass is tied, the new mass only moves down by x/2 whereas the rope extends by x, in this case the energy lost by the new mass (due to change in height) is the same as energy stored in the rope. Your comments on this?

 

[Doc Al]

Alright, I have one alternative explanation too, but I am not sure it is correct
If the rope extends by x, the centre of mass of the rope should therefore move by x/2. Assuming that the center of mass is the point where the mass is tied, the mass only moves down by x/2 whereas the spring extends by x, in this case the energy lost by the mass (due to change in height) is the same as energy stored in the rope. Your comments on this?

The rope is assumed massless: treat it like a massless spring.

Imagine these cases:
(1) Only gravity acts on the mass (no rope or hand): Initial gravitational energy goes into KE of object.
(2) Only gravity and spring force (rope) acts on the mass: Energy goes into spring PE and KE of object
(3) Gravity, rope, and hand all act on the object (slowly lowering it): Energy goes into spring PE and work done on the hand.