Mechanics - Show Forces Are Conservative

In summary, the conversation discusses how to show that two given forces, \vec{F}=c(\vec{\omega}x(\vec{\omega}x\vec{r})) and \vec{F}=d(\vec{\omega}x\vec{v}), are conservative. The equations \vec{\nabla}x\vec{F}=0 and \vec{v}dt=d\vec{r} are mentioned as helpful tools. The first force is identified as the Centrifugal force and the second as the Coriolis force. Suggestions are made to evaluate the x-component and assume symmetry for the first force, and to show that the force along any path is zero for the second force.
  • #1
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Homework Statement


Show that the following forces are conservative:

a) [tex]\vec{F}[/tex]=c([tex]\vec{\omega}[/tex]x([tex]\vec{\omega}[/tex]x[tex]\vec{r}[/tex]))
b) [tex]\vec{F}[/tex]=d([tex]\vec{\omega}[/tex]x[tex]\vec{v}[/tex])

where c and d are constants.

Homework Equations


There are probably more but I think helpful ones include:
[tex]\vec{\nabla}[/tex]x[tex]\vec{F}[/tex]=0
[tex]\vec{v}[/tex]dt=d[tex]\vec{r}[/tex]

The Attempt at a Solution


First I think equation a is just the Centrifugal force in the force equation for a rotating system and equation b is the Coriolis force. I could be wrong though.

We were told for part a that we should just evaluate for the x-component and assume symmetry for the rest. I think for part a the curl should first be calculated.

For part b I believe it would be helpful to show that the force along any path would be zero first. But I'm not sure how to go about this.

Thank you for your time and help!
 
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  • #2
Sorry, this is the wrong thread. Sorry bout that.
 
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1. What is the definition of a conservative force?

A conservative force is a type of force that conserves mechanical energy. This means that the work done by the force on an object is independent of the path taken by the object. The work done by a conservative force only depends on the initial and final positions of the object.

2. How can one determine if a force is conservative?

To determine if a force is conservative, one can use the principle of the conservation of energy. This means that if the total mechanical energy of an object (kinetic energy + potential energy) remains constant, the force acting on the object is conservative.

3. What are some examples of conservative forces?

Some examples of conservative forces include gravitational force, spring force, and electric force. These forces are conservative because their work is independent of the path taken by the object they act upon.

4. How can the work done by a conservative force be calculated?

The work done by a conservative force can be calculated using the formula W = -ΔU, where W is the work done, ΔU is the change in potential energy, and the negative sign indicates that the work is done against the force.

5. What is the significance of conservative forces in mechanics?

Conservative forces are important in mechanics because they allow for the conservation of energy. This means that the total energy of a system remains constant, which is a fundamental principle in physics. Additionally, conservative forces make it easier to analyze and predict the motion of objects in a system.

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