A 33 cm-tall, 6 cm-diameter plastic tube has a sealed bottom. 266 g of lead pellets are poured into the bottom of the tube, whose mass is 45 g, then the tube is lowered into a liquid. The tube floats with 5 cm extending above the surface. What is the density of the liquid?
To find the density of the liquid, we need to use the concept of buoyancy. The upward buoyant force acting on the tube is equal to the weight of the liquid displaced by the tube.
First, let's find the volume of the tube that is submerged in the liquid. The total volume of the tube is calculated using its height and diameter.
Volume of the tube = π * (radius)^2 * height
Given that the diameter of the tube is 6 cm, the radius is half of that, which is 3 cm or 0.03 m. The height of the tube is 33 cm or 0.33 m.
Volume of the tube = π * (0.03 m)^2 * 0.33 m
To find the volume of the cylinder, we can simplify this equation:
Volume of the tube = 0.0309 m^3
Since 5 cm or 0.05 m is the length of the tube that is above the liquid surface, the submerged volume of the tube is:
Submerged volume = Volume of the tube - Volume above the surface
Submerged volume = 0.0309 m^3 - (π * (0.03 m)^2 * 0.05 m)
Next, we need to find the weight of the liquid displaced by the tube. This weight is equal to the buoyant force.
Weight of the liquid = Mass of the liquid * Gravitational acceleration
First, let's find the mass of the liquid. We know that the total mass of the tube and lead pellets is 45 g + 266 g = 311 g or 0.311 kg.
Since the tube floats with 5 cm or 0.05 m above the liquid surface, it is in equilibrium, which means that the weight of the tube and the weight of the displaced liquid are balanced.
The weight of the tube and lead pellets is:
Weight of the tube and lead pellets = Mass of the tube and lead pellets * Gravitational acceleration
Weight of the tube and lead pellets = 0.311 kg * 9.8 m/s^2
Now, since the tube is in equilibrium, the weight of the tube and lead pellets is equal to the weight of the liquid displaced by the tube:
Weight of the tube and lead pellets = Weight of the liquid
0.311 kg * 9.8 m/s^2 = Weight of the liquid
Now we can solve for the weight of the liquid:
Weight of the liquid = 3.0488 N
Finally, we can find the density of the liquid using the equation:
Density = Mass / Volume
Density of the liquid = Weight of the liquid / (Volume of the tube - Volume above the surface)
Density of the liquid = 3.0488 N / (0.0309 m^3 - (π * (0.03 m)^2 * 0.05 m))
By substituting the values and performing the calculation, we can find the density of the liquid.