The gravity of the earth will remain the same, all other things being equal, whether or not the core was hot or cold. Gravity, as Einstein first showed, is a curvature in space-time resulting from mass. Only mass, not temperature, is relevant here.
But what would happen if the liquid metal core of the earth cooled? The magnetic field of the earth would diminish (it is a "dynamo" now, due to the movement of the liquid in the core), as well as stabilize (that is, the north and south poles would stop moving about as they have done for millions of years). We might have some difficulty using common compasses, in this event. Of course, our GPS and celestial navigation would still work fine.
EDIT: The loss of the earths magnetic field would make havoc on earth.
No more magnetism means that the suns solar waves would make atmosphere burn off,( no more AIR!!). The air pressure would diminish from the loss atmosphere. The less air pressure means the boiling point of ALL matter on earth would go down, making it easier for stuff to melt and burn. The oceans would burn off. Earth would be like mars with out an atmosphere. It will happen one day, the earth is heated by radioactive decay in the core. The uranium would eventually be gone. That is why the earth would cool.
This is actually occurring, although at such a slow rate that humans may not even exist as a species by the time it becomes a problem. However, if the Earth's core were to cool significantly, it would stop the formation of new rock through volcanoes and magma flows and also stop the destruction of old rock. This would cause the tectonic plates to stop moving and erosion would eventually level out the surface of the planet.
However, the tectonic plate movement and the rising heat from the Earth's core help support life on the planet. Without it, humans (and a good portion of the rest of the living world as well) would likely go extinct.
it would be cold and it may be less warm
nothing would happen, the sun is constantly going through nuclear reactions
Nothing would happen if the moon was on the left and the sun on the right. The moon and sun are often on different sides of the earth.
we would die
we would all die because it would be to cold
We would not exist. Our sun is a star.
We would freeze to death
Currently the gravitation pull of the Sun is balanced by the momentum the planets have due to their velocities. If the Planets were to slow down they would be pulled closer to the Sun. They could be pulled into the Sun or regain stable orbits as the Sun's gravity increased their speeds again.
Everything not bolted down, including the atmosphere, would fly off into space, the earth would depart from the sun. The sun and all the stars would explode.
nothing would happen, the sun is constantly going through nuclear reactions
Nothing realy really. If it sped up it would spin faster and make shorter days and would revolve around the sun faster. If it slowed down it would spin slower making longer days and it would revolve around the sun slower.
Nothing would happen if the moon was on the left and the sun on the right. The moon and sun are often on different sides of the earth.
The sun would win.
It would melt.
You would be vaporised long before you actually reached the sun.
death
It cannot happen, but if it did we would all die shortly therafter.
you will die