Understanding Energy (kinetic and potential)

[Addem]

I have these two problems on energy where I just don't totally understand the explanations given. One goes:

(With a diagram showing a 14kg mass suspended 5m above the ground by a zero-mass rope, connected to a pulley, where on the other side of the rope is an 8kg mass resting on the ground. The rope is taut before the 14kg mass is released and the vector tension on each points straight up.)

"See the masses in Figure (1) which start out at rest. (i) Find the velocity of the 14kg mass just before it hits the ground. (ii) Find the maximum height reached by the 8kg (and don't worry about hitting the pulley). (iii) Find the fraction of mechanical energy left when the system finally comes to rest."

So with this one I can do the math, I just don't understand the conservation of energy principle here. Every force here is either a gravitational force or a tension force which is itself due to a gravitational force being transferred from one object to another. So I would think, here, that the energy would be conserved.

So like I said, I can do the math and see that the computations imply that the answer to part (iii) is less than 1 but I just don't get why, in light of conservation of energy. Can anyone explain to me why energy is not conserved here?

There is a second question,

"A rocket is launched at escape velocity from the surface of the Earth (radius R). What is its velocity when it is at a distance r from the center of the Earth in terms of G and M(of earth)?"

This one I very much don't understand, because when I read the solution it asserts that initial energy is 0. However, I would think that initial kinetic energy would be mv^2 / 2 and potential energy would be 0.

I've never totally understood energy and I'm not sure if that conceptual fogginess is responsible for why I'm not really getting these two problems. Any help is appreciated.

 

[DimReg]

Well, technically when the mass hits the floor, it's energy is transferred into the floor. Everything is made of atoms, and when two objects collide, those atoms are given more energy individually. In fact, waves would be created in the floor/air, which is responsible for the fact that you would be able to hear it fall.

But really, energy here isn't conserved because you are artificially imposing zero motion, without considering the physical process in which it would occur.

For the second question:

The initial energy is zero, at the surface of the earth.

 

[Andrew Mason]

I have these two problems on energy where I just don't totally understand the explanations given. One goes:

(With a diagram showing a 14kg mass suspended 5m above the ground by a zero-mass rope, connected to a pulley, where on the other side of the rope is an 8kg mass resting on the ground. The rope is taut before the 14kg mass is released and the vector tension on each points straight up.)

"See the masses in Figure (1) which start out at rest. (i) Find the velocity of the 14kg mass just before it hits the ground. (ii) Find the maximum height reached by the 8kg (and don't worry about hitting the pulley). (iii) Find the fraction of mechanical energy left when the system finally comes to rest."

So with this one I can do the math, I just don't understand the conservation of energy principle here. Every force here is either a gravitational force or a tension force which is itself due to a gravitational force being transferred from one object to another. So I would think, here, that the energy would be conserved.

So like I said, I can do the math and see that the computations imply that the answer to part (iii) is less than 1 but I just don't get why, in light of conservation of energy. Can anyone explain to me why energy is not conserved here?

Energy is conserved up to an instant before the 14 kg mass hits the floor. But it is the energy of the system that is conserved, not the energy of each mass individually. Energy is transferred from the 14 kg. to the 8 kg mass.

There is a second question,

"A rocket is launched at escape velocity from the surface of the Earth (radius R). What is its velocity when it is at a distance r from the center of the Earth in terms of G and M(of earth)?"

This one I very much don't understand, because when I read the solution it asserts that initial energy is 0. However, I would think that initial kinetic energy would be mv^2 / 2 and potential energy would be 0.

I've never totally understood energy and I'm not sure if that conceptual fogginess is responsible for why I'm not really getting these two problems. Any help is appreciated.

Potential energy is negative at the surface of the earth. At infinite r, its potential energy is 0. Escape velocity means it has enough kinetic energy at the Earth surface to reach infinite r at which point is speed is 0 (0 KE). Since total energy does not change, its total energy everywhere from R to ∞ is 0.

AM

 

[technician]

The 14kg mass falling through 5m 'gives up' 686J of PE.
This energy appears as PE of the 8kg mass rising by 5m + KE of 8kg mass + KE of the 14kg mass as it hits the ground
This last amount of KE is therefore not part of the total mechanical energy of the masses.

 

[PJC01]

Hello!

I can understand why these concepts might be confusing for you, as energy is a complex and abstract concept. However, I will try my best to explain it in a way that will hopefully clarify things for you.

Firstly, let's talk about the conservation of energy principle. This principle states that energy cannot be created or destroyed, it can only be transferred or converted from one form to another. In simpler terms, the total amount of energy in a closed system remains constant.

In the first problem, the system is initially at rest, with both masses suspended in the air. When the 14kg mass is released, it starts to fall towards the ground due to the force of gravity. As it falls, it gains kinetic energy (energy of motion) and loses potential energy (energy stored in an object due to its position). This is because the mass is moving closer to the ground, thus reducing its potential energy.

At the same time, the 8kg mass on the ground starts to move upwards due to the tension in the rope. This tension is created by the mass of the 14kg mass pulling down on the rope. As the 8kg mass moves upwards, it also gains potential energy and loses kinetic energy. This is because it is moving further away from the ground, thus increasing its potential energy.

Now, let's look at the moment just before the 14kg mass hits the ground. At this point, it has lost all of its potential energy and has converted it into kinetic energy. This means that its velocity will be at its maximum, as all of the energy has been converted into motion. This is why the answer to part (i) is not 0, as you may have initially thought.

Moving on to part (ii), we can see that the 8kg mass has reached a maximum height before it starts to fall back down. At this point, it has lost all of its kinetic energy and has converted it into potential energy. This is why the answer to part (ii) is not 0, as you may have initially thought.

Finally, when the system comes to rest, both masses have lost all of their energy and have converted it into heat and sound. This is why the answer to part (iii) is less than 1, as some energy has been lost in the form of heat and sound.

Now, let's move on to the second problem about the rocket launch. When the rocket is launched, it