A plastic cube having a side length of 10cm floats in fresh water. If the volume of thr cub3 that's submerged in water is 40% of it's total volume, what is the density of the cube?(density of water=1000kg/m³).
We can start by finding the volume of the submerged portion of the cube. Since we know that it is 40% of the total volume, we can set up the following equation:
0.4 x total volume = submerged volume
Let's call the total volume "V". The volume of a cube is calculated by multiplying the length, width, and height. Since all sides have the same length in a cube, we can simplify this to:
V = (side length)^3
Plugging in the given value of 10cm for the side length, we get:
V = 10cm x 10cm x 10cm = 1000cm³
Now we can use this to find the volume of the submerged portion:
0.4 x 1000cm³ = 400cm³
Since the cube is floating in water, we know that its weight is equal to the weight of the water it displaces. This is known as Archimedes' principle. The weight of the water it displaces can be calculated using the density of water and the volume of the submerged portion:
Weight of water displaced = density x volume
Weight of water displaced = 1000kg/m³ x 0.4 x 1000cm³/1000000 = 0.4kg
Since the cube is not sinking or rising, its weight must be equal to the weight of the water it displaces. Let's call the density of the plastic cube "D":
Weight of cube = D x V x density of water
Weight of cube = D x 1000cm³ x 1000kg/m³ = D x 1kg
Setting the weight of the cube equal to the weight of the water displaced, we get:
D x 1kg = 0.4kg
Solving for D, we get:
D = 0.4kg/1kg = 0.4
Therefore, the density of the cube is 0.4 times the density of water, or:
Density of cube = 0.4 x 1000kg/m³ = 400kg/m³
To find the density of the cube, we need to use the concept of buoyancy.
The buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. Since the cube is floating in water, the buoyant force is equal to the weight of the water displaced by the submerged part of the cube.
Given that the submerged volume of the cube is 40% of its total volume, we can calculate the submerged volume as follows:
Submerged Volume = 40% * Total Volume
= 0.4 * (side length)^3
= 0.4 * (10cm)^3
= 0.4 * 1000 cm³
= 400 cm³
Since the cube is floating, the upward buoyant force acting on it is equal to the weight of the water displaced by the submerged volume:
Buoyant Force = Weight of Displaced Water
The weight of the water can be calculated by multiplying the mass of the water by the acceleration due to gravity.
Density is defined as mass divided by volume:
Density = Mass / Volume
Since the weight of an object is equal to its mass multiplied by the acceleration due to gravity, we can rewrite the equation for density as:
Density = (Weight / Gravity) / Volume
Given that the density of water is 1000 kg/m³ (or 1 g/cm³), we can substitute the known values into the equation:
Density = (Weight of Displaced Water / Gravity) / Volume
= (Volume of Displaced Water * Density of Water / Gravity) / Volume
= (400 cm³ * 1 g/cm³ / 9.8 m/s²) / (1000 cm³)
= (400 g / 9.8 m/s²) / (1000 cm³)
= 40 g / (9.8 m/s² * 1000 cm³)
≈ 0.004 kg / (9.8 m/s² * 1000 cm³)
Converting the volume to cubic meters and canceling the units:
Density ≈ 0.004 kg / (9.8 m/s² * 0.001 m³)
≈ 0.004 kg / 0.0098 kg
≈ 0.408 kg/m³
Therefore, the density of the cube is approximately 0.408 kg/m³.