A solid cylinder has a radius of 0.051 m and a height of 0.0030. The cylinder is composed of two different materials with mass densites of 1950 kg/m3. If each of the two materials occupies an equal volume, what is the mass of the cylinder?
Well, well, well, looks like we have a cylindrical puzzle here! Let's see if we can solve it with some humor, shall we?
To find the mass of the cylinder, we need to consider the two different materials it's made of. Since each material occupies an equal volume, it means each material occupies half the cylinder.
Now, if we could just find a way to split that cylinder in half...cue the circus music! 🎪
Let's calculate the volume of the cylinder, V = πr²h. So, V = π(0.051 m)²(0.0030 m).
Now, since we're only dealing with half of that volume for each material, we divide it by 2.
Finally, we can calculate the mass, m = ρV, where ρ is the mass density. Since each material has the same mass density (1950 kg/m³), we just multiply it by the volume we found earlier.
But wait, we haven't considered the units! Remember that π is dimensionless, so it's just a number. After all, a mathematical circus doesn't mess with dimensions!
Now, for the grand finale, let's bring it all together and get that mass!
m = (ρV) + (ρV) = 2ρV
Plug in the values and ta-da! You've got your mass, my friend! Now go celebrate your victory like a clown at a circus party! 🎉
To find the mass of the cylinder, we need to consider the volumes and mass densities of the two materials.
1. Calculate the volume of the cylinder:
- The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
- Substituting the values given, we have V = π(0.051m)^2(0.0030m) = 2.449 × 10^-5 m^3.
2. Divide the volume by 2 to find the volume of one material:
- Since each of the two materials occupies an equal volume, we divide the total volume by 2.
- Volume of one material = (2.449 × 10^-5 m^3) / 2 = 1.225 × 10^-5 m^3.
3. Calculate the mass of one material using its mass density:
- The mass of a material is given by the formula m = ρV, where ρ is the mass density and V is the volume.
- Substituting the values given, we have m1 = (1950 kg/m^3) * (1.225 × 10^-5 m^3) = 2.390625 kg.
4. Since both materials have the same mass, the total mass of the cylinder is twice the mass of one material:
- Total mass of the cylinder = 2 * m1 = 2 * 2.390625 kg = 4.78125 kg.
Therefore, the mass of the cylinder is approximately 4.78125 kg.
To find the mass of the cylinder, we need to first calculate the volume of the cylinder and then multiply it by the average mass density of the two materials.
To calculate the volume of the cylinder, we use the formula for the volume of a cylinder:
V = π * r^2 * h
Where:
V is the volume of the cylinder
r is the radius of the cylinder
h is the height of the cylinder
Given:
radius (r) = 0.051 m
height (h) = 0.0030 m
Substituting these values into the formula, we get:
V = π * (0.051)^2 * 0.0030
Calculating this expression, we find:
V ≈ 1.27 * 10^-5 m^3
Since each of the two materials occupies an equal volume, each material will occupy half of the total volume. Therefore, the volume of each material is:
V_material = (1/2) * V
Substituting the value of V, we find:
V_material ≈ (1/2) * 1.27 * 10^-5
≈ 6.35 * 10^-6 m^3
Now, to find the mass of the cylinder, we multiply the average mass density (ρ) by the volume of each material:
m = ρ * V_material
Given:
mass density (ρ) = 1950 kg/m^3
Substituting the value of V_material and ρ, we get:
m = 1950 kg/m^3 * 6.35 * 10^-6 m^3
Calculating this expression, we find:
m ≈ 0.012 kg
Therefore, the mass of the cylinder is approximately 0.012 kg.