Does the Existence of Dark Matter Contradict the Laws of Physics?

[gaugeboson]

Why is it necessary to assume dark matter exists in the universe? If we feel that galaxies are rotating faster than it is possible for normal matter to do so, they must have less moment of inertia. This will make their rotation fast. Hence, they should have less mass, not more!
I’ve heard that black holes finally evaporate. They then send out all the particles they have sucked in. Thus is it not probable that a cluster of matter would form around an evaporated black hole?
Moment of inertia in a ring is summationMR2. We can consider a spiral galaxy as a ring assuming it has lots of dark matter near its boundaries. Thus M and R will both be greater. Hence moment of inertia should increase, reducing speed of rotation. Thus it supports my counter-dark matter argument.

 

[SpaceTiger]

Why is it necessary to assume dark matter exists in the universe? If we feel that galaxies are rotating faster than it is possible for normal matter to do so, they must have less moment of inertia. This will make their rotation fast. Hence, they should have less mass, not more!

If you imagine a "ring" of matter in a circle around a galaxy's center, the energy (and therefore speed) of that ring's rotation will be determined by the mass interior to it. This comes from Newton's Law of Gravitation.

$$E=\frac{GM_{int}M_{ring}}{r}=\frac{1}{2}I_{ring}\omega^2=\frac{1}{2}M_{ring}^2r^2\omega^2$$

$$\omega=\sqrt{\frac{2GM_{int}}{r^3}}$$

Thus, more mass, faster rotation.

I’ve heard that black holes finally evaporate. They then send out all the particles they have sucked in. Thus is it not probable that a cluster of matter would form around an evaporated black hole?

When black holes evaporate, they tend to emit radiation. Since this will escape the area around the black hole, there will be no cluster left in its wake.