Styrofoam has a density of 300 kg/m^3. What is the maximum mass that can hang without sinking from a 50 cm diameter styrofoam sphere in water? (Assume the volume of the mass is negligible compared to that of the sphere.) V sphere = r^34/3 pie
Calculate the density of sphere whose mass 20g and its diameter 14cm
To find the maximum mass that can hang from a styrofoam sphere without sinking, we need to calculate the buoyant force acting on the sphere in water. The buoyant force is equal to the weight of the water displaced by the sphere.
Given:
Density of Styrofoam (ρ) = 300 kg/m^3
Diameter of Styrofoam sphere (d) = 50 cm
We first need to calculate the volume of the styrofoam sphere:
Radius (r) = d/2 = 50 cm/2 = 0.25 m
Volume of the sphere (V_sphere) = (4/3) * π * r^3
Now we can calculate the buoyant force:
Buoyant Force = Weight of the Water Displaced
Buoyant Force = Volume of Sphere * Density of Water * Acceleration due to Gravity
Density of water (ρ_water) = 1000 kg/m^3 (approximately)
Acceleration due to gravity (g) = 9.8 m/s^2
Buoyant Force = V_sphere * ρ_water * g
Finally, to find the maximum mass (m_max) that can hang from the sphere without sinking, we equate the buoyant force to the weight of the mass:
Weight of the Mass (W_mass) = m_max * g
Setting Buoyant Force = Weight of the Mass, we have:
V_sphere * ρ_water * g = m_max * g
We can cancel out the acceleration due to gravity (g) on both sides and then solve for m_max:
V_sphere * ρ_water = m_max
Substituting the value of V_sphere and ρ_water, we get:
(4/3) * π * (0.25)^3 * 1000 = m_max
Calculate the value of m_max to find the maximum mass that can hang from the sphere without sinking.
To find the maximum mass that can hang without sinking from the styrofoam sphere in water, we need to determine the buoyant force acting on the sphere and compare it to the gravitational force pulling it downwards.
The buoyant force is equal to the weight of the water displaced by the object:
Buoyant force = density of water × volume of water displaced × acceleration due to gravity
For a sphere, the volume can be calculated using the formula you provided:
Volume of sphere = (4/3) × π × radius^3, where radius = diameter/2
Next, we need to calculate the weight of the water displaced by the sphere:
Weight of water displaced = density of water × volume of sphere × acceleration due to gravity
Since the sphere is in equilibrium (not sinking or floating), the buoyant force must equal the gravitational force acting on the sphere:
Buoyant force = Weight of water displaced
Therefore, we can set up the equation:
density of water × volume of water displaced × acceleration due to gravity = density of sphere × volume of sphere × acceleration due to gravity
We can cancel out the acceleration due to gravity on both sides of the equation:
density of water × volume of water displaced = density of sphere × volume of sphere
Substituting the values given:
density of water × volume of water displaced = 300 kg/m^3 × volume of sphere
Now, we can solve for the volume of water displaced:
volume of water displaced = (density of water / density of sphere) × volume of sphere
Substituting the values:
volume of water displaced = (1000 kg/m^3 / 300 kg/m^3) × volume of sphere
Simplifying:
volume of water displaced = 3.333 × volume of sphere
Finally, to find the maximum mass that can hang without sinking, we need to multiply the volume of water displaced by the density of water:
mass of water displaced = density of water × volume of water displaced
mass of water displaced = 1000 kg/m^3 × (3.333 × volume of sphere)
Given the diameter of the sphere is 50 cm, we can calculate the radius (r):
radius = diameter/2 = 50 cm / 2 = 25 cm = 0.25 m
Now we can substitute the value of the radius into the volume of sphere formula you provided:
volume of sphere = (4/3) × π × radius^3 = (4/3) × π × (0.25 m)^3
Calculate the volume of the sphere in cubic meters and substitute it back into the equation for the mass of water displaced:
mass of water displaced = 1000 kg/m^3 × (3.333 × (4/3) × π × (0.25 m)^3)
Finally, you can solve this equation to find the maximum mass that can hang without sinking from the styrofoam sphere in water.
You mean (4/3) pi r^3
r = 0.25 meter
V = .0654 meters^2
mass of water displaced-mass of foam
= (1000-300)(.0654)
= 45.8 kg