Conservation of Mechanical energy

In summary, the conversation discusses the calculation of work required to pull back a hanging section of a chain on a table. The potential energy of each chain link is determined by multiplying the weight of the link with the distance it is pulled. To calculate the total potential energy, an integral is used to represent infinitely small chain segments and their potential energy. The final equation is W = int (mgh dh) from 0 to L/4.
  • #1
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1.A chain on top of a table is stretched out in a straight line perpendicular to the table's edge with one fourth of its length L hanging over the side. How much work is required to pull the hanging section of the chain back into the table. (hint: divide the hanging part into segments of length dy)
2. W = Ui - Uf ; Total energy = U + K ; F(x) = -dU/dx
3. I can't figure out how to set up the integral..
 
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  • #2
An integral is essentially like doing a lot of addition. Think for example that each chain link is 0.05 m long, and weighs 0.1 kg. Assume five links are hanging off the edge.

Think about each chain link is being pulled up discretely. The first link must be moved 0.05m, the second must be moved 0.1m, the third must be moved 0.15m, etc. How much potential energy does each individual link hold?

The integral would essentially represent having links that are infinitely small, but cover that same distance. Does that held at all?
 
  • #3
In other words, what you need is an expression for the potential energy of a chain segment in terms of the displacement of that chain segment (a segment hanging 5 inches down would have displacement of 5 inches).

Now, integrate that over the entire portion of the chain which is displaced.
 
  • #4
So the equation would be int( mgh dh) from zero to h ?
 
  • #5
where h= L/4
 

1. What is the conservation of mechanical energy?

The conservation of mechanical energy is a fundamental principle in physics that states the total amount of mechanical energy in a closed system remains constant, meaning it does not decrease or increase over time. Mechanical energy is the sum of potential and kinetic energy in a system.

2. Why is conservation of mechanical energy important?

The conservation of mechanical energy is important because it allows us to predict the behavior of objects in a given system. It also helps us understand the relationship between different forms of energy and how they can be converted from one form to another.

3. How is mechanical energy conserved?

Mechanical energy is conserved through the principle of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another. In a closed system, the total amount of mechanical energy remains constant as it is converted between potential and kinetic energy.

4. What factors can affect the conservation of mechanical energy?

The conservation of mechanical energy can be affected by external forces, such as friction and air resistance, which can convert mechanical energy into other forms of energy, such as heat. The shape and composition of objects within a system can also impact the conservation of mechanical energy.

5. How is the conservation of mechanical energy applied in real life?

The conservation of mechanical energy is applied in many real-life situations, such as roller coasters, where the potential energy of the initial ascent is converted into kinetic energy as the coaster moves downhill. It is also used in sports, such as a swinging pendulum in a game of golf, where potential energy is transferred into kinetic energy to hit the ball. In engineering, the principle is applied in designing machines and structures to ensure they are energy efficient and safe.

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