# Energy Loss Q: Find Velocity, Acceleration & Mechanical Energy Lost

• deedsy
In summary: Momentum is conserved but there is a change in kinetic energy, soKE = mgL/v^2So the total mechanical energy lost from the rope is mgL/6.
deedsy

## Homework Statement

A (smooth) rope of length L and mass m is placed above a hole in a table. One end of the rope falls through the hole, pulling steadily on the remainder of the rope. Find the velocity of the rope as a function of the distance to the end of the rope, x. Ignore friction of the rope as it unwinds. Then find the acceleration of the falling rope and the mechanical energy lost from the rope as the end of the rope leaves the table. Note that the rope length is less than the height of the table.

## The Attempt at a Solution

Well I already got the right equations for v and a...

v(x) = (2gx/3)^.5
a(x) = [(2g/3)^.5] / 2√x

But I don't understand how to find the mechanical energy loss. It seems like mechanical energy shouldn't be lost here, because the lost PE is just changing into KE...

This is a tricky question that comes up regularly in various guises.
It should state that the rope is inelastic. This means something interesting happens as each new piece is accelerated from rest, apparently instantaneously, to the current speed of the descending part. This constitutes an impulse, so momentum is conserved but not work.

haruspex said:
This is a tricky question that comes up regularly in various guises.
It should state that the rope is inelastic. This means something interesting happens as each new piece is accelerated from rest, apparently instantaneously, to the current speed of the descending part. This constitutes an impulse, so momentum is conserved but not work.

So the change in mechanical energy would be the difference between the sum of all kinetic energies of small sections of the rope and the change in Potential Energy? But how would you know the change in PE if they don't give you a table height?

I got mgL/6 for the sum of all the KE of all small sections of the rope.

deedsy said:
So the change in mechanical energy would be the difference between the sum of all kinetic energies of small sections of the rope and the change in Potential Energy? But how would you know the change in PE if they don't give you a table height?

I got mgL/6 for the sum of all the KE of all small sections of the rope.

Once a section of rope has passed through the hole there will b no further loss of work, so the height does not matter.
Let the length of the rope that has passed through the hole be x and the current speed of that section be v. Consider a small section length dx passing through the hole, going from rest to speed v, and apply conservation of momentum.

1 person

As a scientist, it is important to carefully consider all factors and variables in a scenario. In this case, the mechanical energy loss may not be immediately apparent, but it is still present. While the potential energy of the rope is indeed being converted into kinetic energy, there are other factors at play that result in a net loss of energy.

First, let's consider the rope itself. As it falls through the hole in the table, it is experiencing drag from the air, which will result in a loss of energy. Additionally, as the rope unwinds, there will be some friction between the rope and the edge of the hole, which will also result in a loss of energy.

Furthermore, there may be some external factors at play, such as vibrations or other forces acting on the rope, which could also contribute to energy loss.

Therefore, while the majority of the potential energy of the rope is being converted into kinetic energy, there are other factors that result in a net loss of mechanical energy. In order to accurately calculate the mechanical energy lost, these factors would need to be taken into account.

## 1. What is energy loss?

Energy loss refers to the decrease in the total amount of energy in a system. It can occur through various processes such as friction, heat transfer, and work done by external forces.

## 2. How do you find velocity?

Velocity is a measure of an object's speed and direction of motion. It can be calculated by dividing the distance traveled by the time it took to travel that distance.

## 3. What is acceleration?

Acceleration is the rate at which an object's velocity changes over time. It is calculated by dividing the change in velocity by the change in time.

## 4. How is mechanical energy lost?

Mechanical energy can be lost in a system through various mechanisms such as friction, air resistance, and work done by external forces. This results in a decrease in the total amount of kinetic and potential energy in the system.

## 5. How do you calculate mechanical energy lost?

Mechanical energy lost can be calculated by finding the difference between the initial and final mechanical energies in a system. This can be calculated using the formula: Mechanical Energy Lost = Initial Mechanical Energy - Final Mechanical Energy.

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