could the easterling breach erebor?
 

Now,in the northern theather of th wotr,erebor was besiege by the easterling,but they were unable to breach the mountain until they hear that sauron was defeated and the siege was broken.if the easterling has much more time,could they breach erebor and finally defeat the dwarven kingdom once and for all?

The reason the Easterlings were even near Erebor was because of Sauron. Like most Humans, the Easterlings were most susceptible to tribalism and inter-tribe squabbling and war, taking land and riches from each other until a powerful lord (like Genghis Khan) could establish hegemony and unite all the tribes into a single army. Sauron could be that powerful lord. In its absence, the Easterlings would "melt away" back to their own lands and take up their squabbles where they had left them.

The siege,in this scenario,happen way before the ring destruction.with so much time,who knows.

A siege is by nature a battle plan that takes months to see to its end. So it may well have been begun long before the destruction of the ring, but ended with it. As to why the Easterlings were not able to extricate the Dwarves before the ring was cast into the flames, I think that has to do with both the nature of siege and Dwarves, the heartiest and stubbornest of the Children of Eru.

Do easterling have catapults?could they bring down the side of the lonely mountain with tyat?or at least make a hole in the gate?

It took three days for the Easterlings to overrun Dale, but the Dwarves were "the most redoubtable warriors of all the Speaking Peoples" ("Of Dwarves and Men"). The Easterlings were presumably numerous, since they were able to defeat the Dwarves and their Dalish allies who were probably more formidable as soldiers and better equipped.

It was not a matter of combat but of morale which weakened the Easterlings and brought victory to the Dwarves and Men of Dale, which implies that the siege could have been maintained in different circumstances. That being said, surely beseiging Erebor would be an utterly monumental task - an entire mountain with tunnels and halls and, I would imagine, a very great store of weapons and armour.

Could reinforcements have come from the Iron Hills? We hear little of them by this late in the Age. There is no mention, for instance, that any force attacked them, yet before Erebor and Dale were rebuilt Gandalf was concerned that "To resist any force that Sauron might send to regain the northern passes in the mountains and the old lands of Angmar there were only the Dwarves of the Iron Hills." So at one time the Iron Hills were considered an Eastern defence - did Erebor and Dale replace this? As such it's difficult to know if the Dwarves could receive reinforcements. That being said, perhaps Thranduil could have come to their aid. It depends how threatening Dol Guldur was at the time, I suppose.

As has been stated in numerous other recent threads about the Easterlings, we know next to nothing about their military equipment or technology. I suppose they could potentially have been afforded siege engines by Sauron, but given that their military effectiveness at the time was limited by Sauron's inability to send reinforcements from Dol Guldur it's hard to say how much support they had been given.

Isnt iron hills dont have a sizable number of dwarves anymore since the refounding of erebor?im pretty sure that it has become obscure since erebor was refounded.as for catapult,they could get it from sauron,and that catapult could add some firepower,because its able to throw high enough to bypass minas tirith's first wall,which is very high,while at the same time remain out of range from gondorian machine in the wall.

There's no definitive statement either way, but as I say given that we have no real mention of the Iron Hills playing any role in the affairs of the last years of the Third Age it seems not unlikely that they were at least depopulated after the reclamation of Erebor. It might be reasonable to imagine that the Dwarves may have kept their mining operations running there, as they did in the Blue Mountains, but it doesn't seem especially likely that they kept a substantial settlement there after Dáin became King Under the Mountain, certainly not enough that could relieve a siege. The Dwarves' numbers were dwindling by that time at any rate.

As I said. Erebor is a mountain, however, not just a walled city, so it's debatable as to how effective a catapult would be regardless of its trajectory. There'd be nothing to fling missiles over and one could hardly expect that they could beat the walls of the mountain down. The Easterlings "could not take the Gate" so it seems that beyond that there were few if any major exploitable points on the exterior of the Mountain.
Again, they could have received siege engines from Sauron, but we have no proof that they did, and given that Sauron's engines seem to have primarily been manufactured in Mordor it's unconvincing that they had access to them. This Easterling army seems to have derived from northern Rhûn, distantly from Mordor, and they did not receive aid from Dol Guldur because of Thranduil (assuming Dol Guldur, in the middle of the forest, even had any storage of such artillery), so their access to advanced engines seems not especially likely to me. If they couldn't take the gate, whatever artillery they had must not have been very effective, or the defences were simply too strong.
The Easterlings' main aim seems to have been to capture the gate, presumably by storming it with Men and perhaps with rams and ladders, but their main recourse like so many ancient sieges would probably be to starve the enemy out. I think had the Easterlings not had cause to waver, the siege would have been protracted and bloody with no clear outcome unless either side could bring reinforcements to their succour.

A siege of an entire mountain (Erebor) brings to mind the 8-month Crusader siege of Antioch during the First Crusade. There werne't enough men to meffectively close off even the city, a mountain, with dwarf engineering technology able to send out scouting groups, raiding/harassing parties etc. at their own initiative would be highly difficult (at least, without a dragon patrolling around Erebor :) ). There would presumably be enormous food and storage prepared (again, not unlike castle preparations in the Middle east with large water retaining stone pits). And even Thorin and Company were prepared to put up resistance on a hastily build wall against the combined elf/man army.