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Initially taking a beaker containing water of having total mass of water and beaker as W. Now when an external object of weight w' is submerged inside the beaker containing water having density d, will the net weight of the beaker will be (w +w') or something like (w+ dVg) if its this then how can it be explained ?

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There is a buoyant force pushing upwards on the submerged object, which is equal to the weight of the water that is displaced. If the submerged object is suspended from a string, and doesn't touch the bottom of the beaker, the weight of the beaker will increase by the magnitude of the buoyant force, because an upwards buoyant force on the object requires an equal magnitude force directed downward via Newton's 3rd law. Because there is an upwards force on the submerged object, it will have an apparent weight that is less than its true weight by the amount of the buoyant force.

Assuming that the submerged object sinks all the way to the bottom of the flask (i.e, it's not suspended by a string), the beaker weight will increase by the buoyant force on that object plus the apparent weight of the submerged object if it was supported by a string. In other words, the beaker weight will increase by the total weight of the submerged object.

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