A non rotating asteroid of huge mass collides with earth. After the collision, would the length of the day on earth increase or decrease? Why?
There is insufficient information to give an answer. There are two effects in play here:
The bigger one depends on the impact parameter -- basically how off-center the impact is. If you imagine the impactor coming in in the equatorial plane, due to the Earth's rotation, one side of the Earth (as viewed from the impactor) is approaching it and one side receding. Unless it hits straight-on, it has angular momentum relative to the Earth and the impactor's angular momentum will remain with the combined body.
So if it hits on the approaching side it's angular momentum will be oppositely-directed to the Earth's and the final body will have less angular momentum and the day will be longer.
If it his on the receding side, it's angular momentum will be directed the same as the Earth's and the final body will have more angular momentum than the Earth does now and the day will be shorter.
If it his dead-on-center, the combined body will have the same angular momentum as the Earth now does.
The second effect is due to the changed moment of inertia of the final body compared to the Earth's. The moment of inertia of a sphere depends on its mass (all else being equal, the moment of inertia is proportional to the mass), the distribution of mass (a body with a denser core has a lower moment of inertia than one of the same mass which is uniform in density), and the size (for the same mass, the larger body has the higher moment of inertia.)
Because of that, if the collision leaves the combined body's angular momentum the same as the Earth's, the result will depend on how the resultant body compares with Earth.
It will have a larger mass, which will increase the moment of inertia. It may be larger (which also increases the moment of inertia), but it also may be smaller if the added mass actually winds up compacting the Earth's material to higher density. (I think this is a long-shot, but it's certainly possible.) If the impactor is extremely dense, it will sink to the center of the combined body and will have little effect on the final moment of inertia.
The combination of these moment of inertia effects will probably more often be a higher moment of inertia and thus a slower rotation rate.
But this effect is probably dominant only for a head-on collision. If the collusion is it all glancing, the final body will have substantially different angular momentum than the Earth presently has and the rotation rate change will be mostly controlled by that.
Bottom line: To answer this -- even to give you the sign of the change -- you need to specify exactly how the asteroid is approaching the Earth (what direction, what speed) its mass, and its composition (though this last only matters if it's pretty exotic.)
On average for "normal" asteroids the effect is probably a small lengthening of the day since the first effect should average out to close to zero, while for normal materials the second effect will probably mostly increase the day's length. But that's the average of a big range in results from a significant shortening of the day to a significant lengthening of it.