The word to notice in the question is "cut" not smashed, cracked open, exploded, vaporized, or ripped apart.
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If a planet was cut into big pieces, gravity would hold it together and it will be drifting around space together. Imagine it like a big group of perfectly sliced asteroids floating in space orbiting the sun due to the sun's gravitational pull. Some slices may separate from the group if another force acts upon it. Such as another asteroid colliding it and setting it off course. Other than that, they are not so different from asteroids except their physical features of being sliced.
If a planet was cut into small/tiny pieces, then all these pieces would end up separating because of the mass each piece holds. Everything has a gravitational pull toward it due to it's mass. The more mass an object has, the stronger the gravitational pull. The smaller mass an object has, the weaker the gravitational pull. With this in mind and the idea that the planet is cut into small/tiny pieces, they will roam aimlessly in space because their gravitational pull toward each other isn't strong enough to hold it together. It will wander aimlessly because of other gravitational forces such as other the sun, planets, moons, and even an asteroid passing through these small cut-like pieces will move it in random directions.
If a planet was cut into small/tiny pieces, then all these pieces would end up separating because of the mass each piece holds. Everything has a gravitational pull toward it due to it's mass. The more mass an object has, the stronger the gravitational pull. The smaller mass an object has, the weaker the gravitational pull. With this in mind and the idea that the planet is cut into small/tiny pieces, they will roam aimlessly in space because their gravitational pull toward each other isn't strong enough to hold it together. It will wander aimlessly because of other gravitational forces such as other the sun, planets, moons, and even an asteroid passing through these small cut-like pieces will move it in random directions.
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What you choose as a best answer is incorrect. The resulting chunks would not 'float' around with some separation. Campbelp2002 is right, gravity would hold everything together (very tightly) as if nothing had been cut. The pieces would be 'crushed' back together tightly by gravity
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Well, yes, gravity would still hold it in its spherical shape, cutting wouldn't change that. You are basically talking about creating an instant fault line that extends through the Earth. This would almost certainly trigger at least some seismic and volcanic activity as this new fault appeared in parts of the crust under stress due to plate tectonics. Depending on where and how and how many cuts were made, it could be minor to globe changing severe. Really kinda depends on what you mean by "cut."
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A drop of water has a greater tensile strength relative to its size than the Earth does. As has been pointed out, there would be local earthquakes, and big ones at that, but the Earth as a whole wouldn't notice.
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Gravity would hold it together as if it had never been cut.