The ratio of crown's apparent weight to its actual weight is (1-[tex]\frac{ρ_{w} }{ρ_{c} }[/tex].
Let then density of the crown is ρ_{c}, the density of the water is ρ_{w}. and the volume of the crown is V
The actual weight of the crown is given as
Wa=ρ_{c}×V×g
Since the volume of the water displaced by the crown is equal to the volume of crown, therefore the buoyancy force exert by the water on the crown is given as
Fw=ρ_{w}×V×g
The buoyancy force will act in the opposite direction of the weight of the crown, therefore the apparent weight of the crown is
Wapp=[ρ_{c}-ρ_{w}]×V×g
Now the ratio of the apparent weight to the actual weight is
=[tex]\frac{[ρ_{c}-ρ_{w}]×V×g}{ρ_{c}×V×g}[/tex]= (1-[tex]\frac{ρ_{w} }{ρ_{c} }[/tex]
Answer:
[tex]\frac{W_{ap} }{W_{ac} } = 1-\frac{rho_w}{rho_c}[/tex]
Explanation:
Assuming crown has volume V, the actual weight of crown
[tex]W_{ac}[/tex] is expressed in terms of magnitude of the acceleration due to gravity, therefore,
[tex]W_{ac} = rho_{c}*gV[/tex]
Also, assuming the crown has volume V, and taking water density to be [tex]rho_w[/tex]
The apparent weight [tex]W_{ap}[/tex] of the crown submerged in water expressed in terms of the acceleration due to gravity
[tex]W_{ap} = (rho_c - rho_w)*gV[/tex]
Finally, the ratio of crown's apparent weight (Wap) to crown's actual weight is given by;
[tex]\frac{W_{ap} }{W_{ac} } = 1-\frac{rho_w}{rho_c}[/tex]
