Total energy of system along axis

In summary, the total energy of an isolated system is given by E = \frac{1}{2}mv^{2} + \frac{1}{2}Iω^{2} + mgh.
  • #1
alterecho
19
0
Its stated that total energy of an isolated system is given by
E = [itex]\frac{1}{2}[/itex]m[itex]v^{2}[/itex] + [itex]\frac{1}{2}[/itex]I[itex]ω^{2}[/itex] + mgh.

Whats the correct form when the velocity is along 2 axis, that is, along x and y-axis like v = (2, 1)?
Should the resultant be taken or just add the kinetic energies along each axis?

I've been searching awhile but can't find an accurate answer.

thanks in advance.
 
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  • #2
alterecho said:
Whats the correct form when the velocity is along 2 axis, that is, along x and y-axis like v = (2, 1)?
Should the resultant be taken or just add the kinetic energies along each axis?
The translational KE would be:
[tex]KE = 1/2mv^2 = 1/2m(v_x^2 + v_y^2)[/tex]
 
  • #3
Thanks. I have a further related question. Now suppose i want to extract the final velocity given the initial total energy, the final rotational energy (assuming the potential energies are 0 in this case) using the law of conservation of energy given by,

[itex]\frac{1}{2}[/itex]m[itex]u^{2}[/itex] + [itex]\frac{1}{2}[/itex]I[itex]ω^{2}[/itex] = [itex]\frac{1}{2}[/itex]m[itex]v^{2}[/itex] + [itex]\frac{1}{2}[/itex]I[itex]W^{2}[/itex]

or

m[itex]u^{2}[/itex] + I[itex]ω^{2}[/itex] = m[itex]v^{2}[/itex] + I[itex]W^{2}[/itex]

since initial energy of the system is known, left side of the equation becomes [itex]E_{i}[/itex],
[itex]E_{i}[/itex] = m[itex]v^{2}[/itex] + I[itex]W^{2}[/itex]

since m, I and [itex]W_{2}[/itex] are known, i get the equation for finding the velocity as,
v = [itex]\sqrt{\frac{E_{i} - IW^{2}}{m}}[/itex]

But how do i break it down into x and y coordinates?
 
  • #4
alterecho said:
But how do i break it down into x and y coordinates?
There's no way to tell the final velocity direction from conservation of energy alone. You'll have to know the details of the constraint forces involved. (Energy is a scalar, not a vector.)
 
  • #5
Oh. But I've read that for resolution of collision, they take into consideration energy conservation. So how would they resolve it? Are we supposed to know the initial and final velocities and then calculate the final angular velocity from that?
 

What is the definition of total energy of a system along a given axis?

The total energy of a system along a given axis is the sum of all forms of energy present in the system, including potential and kinetic energy. It represents the overall energy of the system in a specific direction.

How is the total energy of a system along an axis calculated?

The total energy of a system along an axis is calculated by adding the potential energy and kinetic energy along that axis. The potential energy can be calculated using the formula U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. The kinetic energy can be calculated using the formula K = ½mv², where m is the mass and v is the velocity.

What is the relationship between the total energy of a system and its position along an axis?

The total energy of a system varies depending on its position along an axis. As the position changes, the potential and kinetic energies also change, resulting in a different total energy. This relationship is described by the conservation of energy principle, which states that energy cannot be created or destroyed, only transferred between different forms.

How does the total energy of a system change over time along an axis?

The total energy of a system may change over time along an axis due to the transfer of energy between potential and kinetic forms. For example, as an object falls and gains speed, its potential energy decreases while its kinetic energy increases. However, the total energy of the system will remain constant as long as no external forces act upon it.

Why is the concept of total energy along an axis important in scientific studies?

The concept of total energy along an axis is important in scientific studies because it helps us understand and analyze the behavior of systems. By considering the total energy of a system, we can predict how it will change over time and how it will interact with other systems. This concept is also crucial in fields such as mechanics, thermodynamics, and electromagnetism.

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