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User:Manishearth/Incubator

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To see my major contributions, go to User:Manishearth/Bigcontribs

Working on a script, list of script pages at User:Manishearth/GadgetUS


DO STUFF BELOW THIS LINE PLEASE




(Just for the latex syntax)

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where r is radial distance of pont in polar coordinates, is unit tangential vector, R is radius of solenoid

Rewrite of Reduced Mass

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Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not reduced. In the computation one mass can be replaced by the reduced mass, if this is compensated by replacing the other mass by the sum of both mass.

Uses

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Given two bodies, one with mass and the other with mass , they will orbit the barycenter of the two bodies. The equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass where the force on this mass is given by the gravitational force between the two bodies. The reduced mass is always less than or equal to the mass of each body and is half of the harmonic mean of the two masses.

In such a two body problem, the moment of inertia of the system about the center of mass is given by:

Where r is the distance between the two bodies.

The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. This relationship is determined by the physical properties of the elements as well as the continuity equation linking them.

Other uses

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Reduced mass crops up in a multitude of two-body problems. In a collision with a coefficient of restitution e, the change in kinetic energy can be written as , where vrel is the relative velocity of the bnodies before colission.

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Todo

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  • Restructure
  • Derive moment of inertia of two bodies = mu*r^2, where r is dist betwn two bodies
  • Add more "other uses"
  • It does not only pertain to two body problem, it crops up in all two-body interactions.
  • Images
  • Explain spring connection with halfmuvsquared formula.

Maxwell (Not to be published, just using WP's LaTeX)

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Starting conditions:


Verifying our choice of B with ,

Now, using



Therefore, E is independant of time.

Now, by
Let


Now, by symmetry of the original conditions, we can assume that:
at a point (x,y,z)
And also,
, by z-component of (1)






By x and y components of (1),

For simplicity, we can assume:




(Where l,m,n,a,b are constants)





Applying to ,






This answer can be verified with the four Maxwell's Laws.