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Styrofoam has a density of 150kg/m3. What is the maximum mass that can hang without sinking from a 50-cm-diameterStyrofoam sphere in water ?Assume the volume of the mass is negligible compared to that of the sphere.

Short Answer

Expert verified

Maximum mass that can hang without sinking from Styrofoam sphere is 56kg.

Step by step solution

01

Definition of Styrofoam 

Closed-cell extruded polystyrene foam, or XPS, is a trademarked brand of Styrofoam. Blue Board is another name for this foam, which is used for building insulation, thermal insulation, and water barriers.

The polymer polystyrene, with styrene as the repeating monomer, is used to make Styrofoam. Only carbon and hydrogen make up styrene.

02

Expressions for force and mass

Use the expression for buoyant force to determine the required mass.

The buoyant force is expressed as follows:

FB=ρfVfg

Here, ρfis density of the fluid andvf is volume of displaced fluid which is equal to volume of Styrofoam sphere.

For the mass that can hang without sinking, the downward gravitational force must be balanced by the upward buoyant force.

In this case, the net force on the mass is expressed as follows:

Fnet=FB-msg

Here, FBis buoyant force msis mass of Styrofoam sphere, andgis acceleration due to gravity.

03

Calculation of mass

Combine the above two relations.

mg=ρfVfg-msg

Here, mis mass of the object.

Volume of Styrofoam sphere is,

Vf=43πr3

Mass of the Styrofoam sphere is,

ms=ρsvs

Here, ρsis density of Styrofoam sphere.

Replace mswith ρsvfand vfwith 43πr3in mg=ρfVfg-msg.

m=ρf43πr3-ρs43πr3

Substitute 1000kg/m3for ρf,25cmfor r,150kg/m3for ρsm=1000kg/m3-150kg/m343π(25cm)31×10-2m1cm3

=55.6kg

Maximum mass is56kg

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