Q:

If Earth is tilted, why is Polaris always above the same spot?

Why is Polaris, the North Star, always above (or near) the North Pole? If Earth is tilted, Polaris' path should be in winter 23 degrees away from its path in summer, or not?

A:

You are correct that the axis of the Earth's rotation is tilted with respect to the plane of its orbit by 23 degrees. But it is incorrect that the direction that the axis points changes by a large amount (it should be 2*23 degrees) over a 6 month time span.

Your assumption: If axis it pointed at Polaris at (1), then it should be pointed at a different star at (2).
The axis remains pointed in the same direction throughout the entire year because the laws of physics are that the axis of a spinning object remains pointed in the same direction unless a torque acts on the body to change its orientation. There are torques acting on the Earth (namely the Sun and Moon), but it takes 13000 years to change the direction from "one way to the opposite way", not 6 months.

Correct: The orientation of the Earth's spin axis remains pointed at the same star throughout the year (ignoring the slow, 26000 year precession cycle).

A:

You may hear about something called the celestial sphere. While we now know the Earth is rotating on its axis, to ancient people it appeared as though the stars were rotating around us (and they are according to our own frame of reference). This rotating globe of stars can itself be mapped on a globe in the same fashion that we can map the Earth's land and water.

There are a few reasons why Polaris is always seen in the same position on this celestial sphere.

As the globe of the Earth and the celestial sphere rotate relative to each other, they share an axis. This center of rotation does not move.
While some planets (like Mars) rotate on an axis that is itself wobbling relative to the ecliptic, Earth has a stable axis thanks to our large moon. This axis always points in the same direction and so it's always aligned with the same stars in our galaxy.
The stars in our galaxy are orbiting the galactic center very slowly from our perspective, so human civilization would not have observed any significant change in the stars' positions across its lifetime.
During the Earth's orbit around the Sun, its own change in position is so slight relative to the galactic scale that no significant stellar parallax can be expected.

A:

Kick (or toss) an AFL/rugby football (a prolate spheroid) so that it spins rapidly (around its long axis) at the same as it traverses an arc (falling under gravity). You should be able to observe that the spin axis is parallel transported (at least if the perturbation by air resistance is weak enough to neglect).
The earth is spinning. To change the direction it spins in (with respect to the inertial frame of distant stars) it would be necessary to apply some torque to the earth. A sun's gravity does not apply any torque to a spherical planet (due to the symmetries of the Newtonian forces applied).
Since the earth is not quite a perfect sphere (e.g. is slightly oblate) there is a small torque applied by the sun's gravitational field gradient, so over millennia the planet's north axis gradually precesses to point toward different constellations.

