An iron wrench (density = 7800 kg/m3) feels 3N lighter in water. (a) What is the mass of the wrench? (b) How would your answer change if it took a denser fluid to make the wrench feel 3N lighter?

##”(a)”## The answer is ##2387.8 “kg”##
Method
From Archimedes’ Principle: the upthrust force on an object placed in a fluid is equal to the weight of the fluid displaced
and volume of the fluid displaced is equal to the volume of the object.
where ##W## is the real weight of the wrench and ##W’## is the apparent weight due to the Upthrust force(##U##).
From the formula: ##”density” = “mass”/”volume”##
##=>## The volume of water displaced, ##V_w= m_w/rho_w##
Now, our upthrust, ##U=”weight of water displaced”(m_wg)=3##
##=> m_wg=3##
##=>m_w=3/g##
##=>V_w=3/(grho_w)##
Similarly, volume of wrench ##V=m/rho##
Equate the two volumes to get the only unknown; mass
##=>m/rho=3/(grho_w)##
##=>m=(3rho)/(grho_w)##
##=>m=(3xx7800)/(9.8xx1)=2387.8 “kg”##
##”(b)”## If the fluid is denser then, the mass would be smaller
I know this from ##m=(3rho)/(grho_w)##
That is, ##m ## ## alpha ## ##1/rho_w##