Hey guys - this problem is pretty tough for me...any help? Don't know what to do....
In seawater, a life preserver with a volume of 4.00×10^−2 m^3 will support a 76.0 kg person (average density 980 kg/m^3), with 20.0 % of the person's volume above water when the life preserver is fully submerged. What is the density of the material composing the life preserver? (answer in kg/m^3)
Thanks!
weight displaced salt water= bouyancy force
Volume displace*densityseawater*g=bouyant force
(4E-2m^3*density*g+.8*76/980*g)= 76g+
4*E-2*density*g
check that and solve for density. Notice the g factor divides out.
Of course, I can help you with that! To find the density of the material composing the life preserver, we need to use the concept of buoyancy.
Buoyancy is the upward force exerted on an object immersed in a fluid (in this case, seawater), which counteracts the force of gravity.
We're given that the life preserver has a volume of 4.00×10^−2 m^3 and that it can support a 76.0 kg person with 20.0% of their volume above water when it is fully submerged.
To start solving this problem, we need to determine the volume of the person that is submerged in water. We can do this by multiplying the total volume of the person (which is their weight divided by the average density) by the percentage submerged.
Volume submerged = Volume of the person × % submerged
= (Weight of the person / Average density) × % submerged
Calculating the volume submerged:
Volume submerged = (76.0 kg / 980 kg/m^3) × 0.20
= 0.0155 m^3
Now, let's find the volume of the life preserver below water. Since the life preserver supports the person, the upward buoyant force on the life preserver is equal to the weight of the submerged person.
Buoyant force = Weight of the submerged person = mass of the submerged person × acceleration due to gravity
= (Volume submerged × Average density) × acceleration due to gravity
Using the known values:
Buoyant force = (0.0155 m^3 × 980 kg/m^3) × 9.8 m/s^2
= 143.414 N
The buoyant force can also be calculated using Archimedes' principle:
Buoyant force = Weight of the life preserver
= Volume of the life preserver × Density of the life preserver × acceleration due to gravity
We're trying to find the density of the material composing the life preserver, so let's express the formula in terms of density:
Density of the life preserver = (Weight of the life preserver) / (Volume of the life preserver)
= Buoyant force / (Volume of the life preserver)
Substituting the values we know:
Density of the life preserver = 143.414 N / 4.00×10^−2 m^3
= 3585.35 kg/m^3
Therefore, the density of the material composing the life preserver is 3585.35 kg/m^3.