Is it true that the further a planet is from the Sun, the faster it rotates around itself?
There is no connection between a planet's distance from the Sun and its rate of rotation (spin on its own axis or Planetary Rotation Period). While it is true that, generally speaking, the gas giants have higher rotation rates than the earth like planets closer to the Sun, any correlation is only coincidental. Mars, for example, is further from the Sun than Earth but has a slightly slower rate of rotation - its planetary rotation period is 1.03 Earth days. Pluto. the furthest planet from the Sun, has a planetary rotation period of 6.39 Earth days. You can see in the plot below that there is no correlation between the planetary rotation period and the distance from the Sun.
There is, however, a relationship between a planet's distance from the Sun and its period of revolution. Kepler's third law of planetary motion says that the square of the planet's orbital period is proportional to the cube of its semimajor axis. For now, just read that as saying
that a planet's 'year' is determined ONLY by its average distance from the Sun. The further away from the Sun it is, the slower the planet's orbital speed and the longer its path.
Both of those factors result in taking longer to make one complete orbit and a planet having a longer year.
Paul Walorski, B.A., Part-time Physics Instructor
'The mathematician's patterns, like the painter's or the poets, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.'