Styrofoam has a density of 300kg/m3. What is the maximum mass that can hang without sinking from a 40.0cm -diameter Styrofoam sphere in water?

Please help!

V = (4/3) pi r^3

V = (4/3) pi (.2)^3 = .0335 meters^3

mass of ball = 300*.0338 = 10 kg

mass of water displaced =1000*.0335 = 33.5 kg

so Archimedes can hold up 33.5-10 = 23.5 kg

To find the maximum mass that a Styrofoam sphere can support without sinking, we need to consider the buoyant force acting on it in water.

The buoyant force is equal to the weight of the water displaced by the submerged volume of the sphere.

First, we need to calculate the volume of the submerged sphere.

Given:
Density of water, ρwater = 1000 kg/m^3
Diameter of the sphere, d = 40.0 cm = 0.40 m
Radius of the sphere, r = d/2 = 0.40/2 m = 0.20 m

The volume of the submerged sphere, Vsub = (4/3) * π * r^3

Vsub = (4/3) * 3.14 * (0.20)^3
Vsub = 0.0335 m^3

Now, we can calculate the buoyant force acting on the sphere.

Buoyant Force, Fbuoyant = ρwater * Vsub * g

where g is the acceleration due to gravity, g = 9.8 m/s^2

Fbuoyant = 1000 * 0.0335 * 9.8
Fbuoyant = 327.8 N

The buoyant force acts upward and is equal to the weight of the sphere, which is equal to the mass of the sphere (m) multiplied by the acceleration due to gravity (g).

Fbuoyant = m * g

Rearranging the equation, we can find the maximum mass that the Styrofoam sphere can hold without sinking:

m = Fbuoyant / g

m = 327.8 / 9.8
m = 33.5 kg

Therefore, the maximum mass that can hang without sinking from a 40.0 cm diameter Styrofoam sphere in water is 33.5 kg.

To solve this problem, we'll need to use Archimedes' Principle. The principle states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces.

Step 1: Calculate the volume of the Styrofoam sphere:
The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere. Since you have the diameter of the sphere, you can calculate the radius by dividing the diameter by 2.

Given:
Diameter = 40.0 cm

Calculations:
Radius = Diameter / 2 = 40.0 cm / 2 = 20.0 cm = 0.2 m

Volume = (4/3)πr³ = (4/3)π(0.2 m)³
Volume = (4/3)π(0.008 m³) ≈ 0.03351 m³

Step 2: Calculate the weight of the water displaced:
To get the weight of the water displaced, we need to multiply the volume of the Styrofoam sphere by the density of water (ρ_water).

Given:
Density of water (ρ_water) = 1000 kg/m³ (standard value for water)

Weight of water displaced = ρ_water * Volume

Calculations:
Weight of water displaced = 1000 kg/m³ * 0.03351 m³
Weight of water displaced ≈ 33.51 kg

Step 3: Calculate the maximum mass that can hang without sinking:
The maximum mass that can hang without sinking is equal to the weight of the water displaced, as per Archimedes' Principle.

Maximum mass = Weight of water displaced ≈ 33.51 kg

Therefore, the maximum mass that can hang without sinking from a 40.0 cm diameter Styrofoam sphere in water is approximately 33.51 kg.