Thank you for your reply
@kuruman! That is very kind to offer. Yes I am interested, here it is:
For an object immersed in water the net force is
##\vec {F_{net}} = B\hat j - mg\hat j##
##F_{net} \hat j = B\hat j - mg\hat j##
##F_{net} \hat j = (B - mg)\hat j## (Factoring out j-hat unit vector to and then cancelling them)
##F_{net} = B - mg ##
##F_{net} = ρ_{fluid}V_{fluid}g - ρ_{object}V_{object}g## (From Archimedes principle that states that the buoyancy force is equal to the weight of the displaced water)
Note: I let ##V_{fluid}## represent the volume of fluid displaced
Now we take special cases of our expression,
##F_{net} = ρ_{fluid}V_{fluid}g - ρ_{object}V_{object}g##
The first case we will consider ##ρ_{object} > ρ_{fluid}##, and consider the object displacing the maximum amount of water to get the max ##B##, therefore ##V_{fluid} = V_{object}##
##F_{net} = ρ_{fluid}V_{object}g - ρ_{object}V_{object}g##
##F_{net} = V_{object}g(ρ_{fluid} - ρ_{object})##
And since ##ρ_{object} > ρ_{fluid}## then ##ρ_{fluid} - ρ_{object} < 0## so ##F_{net} < 0## which means that the object has a net force downward and will sink
For the second case we will consider ##ρ_{object} < ρ_{fluid}##, and consider the object displacing the maximum amount of water to get the max ##B##, therefore ##V_{fluid} = V_{object}##
##F_{net} = ρ_{fluid}V_{object}g - ρ_{object}V_{object}g##
##F_{net} = V_{object}g(ρ_{fluid} - ρ_{object})##
And since ##ρ_{object} < ρ_{fluid}## then ##ρ_{fluid} - ρ_{object} > 0## so ##F_{net} > 0## which means that the object has a net force upwards and will float.
For the third case we will is ##ρ_{object} = ρ_{fluid}##, and consider the object displacing the maximum amount of water to get the max ##B##, therefore ##V_{fluid} = V_{object}##
##F_{net} = ρ_{fluid}V_{object}g - ρ_{object}V_{object}g##
##F_{net} = V_{object}g(ρ_{fluid} - ρ_{object})##
And since ##ρ_{object} = ρ_{fluid}## then ##ρ_{fluid} - ρ_{object} = 0## so ##F_{net} = 0## which means that the object is in vertical equilibrium so if placed gently then it the object will float. However, if put into the water with an initial velocity it will remain sinking (Assuming no fluid resistance) in accordance with Newton First Law.
I think the main think I learnt from this was to consider the most extreme case where ##B = B_{max}## since this is the maximum amount of upwards force from the fluid that the object can get.
Is this please all correct?Many thanks!