a cube of side L is made of a substance that is 1/4 as dense as water. WHen placed in a calm water bath the cube will a) float with 1/2 L abovethe surface b)sink to the bottom, c) float with 1/4 L above the surface d) float with 1/4 L below the surface e) float with 3�ã1/4
Only 1/4 of the cube must be submerged to support the weight (Archimedes).
To determine what will happen to the cube when placed in a calm water bath, we need to consider the density of the cube and the density of water.
The density of a substance is defined as its mass per unit volume. Since the cube is made of a substance that is 1/4 as dense as water, it means that its density is 1/4 times the density of water.
Now, let's consider the scenarios:
a) If the cube had the same density as water, it would have neither buoyancy nor weight, and it would stay fully submerged. Therefore, the cube will not float with 1/2 L above the surface.
b) Since the cube has a lower density than water, it will experience upward buoyant force. The buoyant force is equal to the weight of the water displaced by the cube. Since the cube is less dense than water, it will displace a volume of water that is greater than its own volume, hence the buoyant force will be greater than the weight of the cube. Therefore, the cube will float and not sink to the bottom.
c) and d) If the cube floats, it will displace a volume of water equal to its own volume. Since the cube is made of a substance that is 1/4 as dense as water, its volume is 4 times greater than the volume of water it displaces. Therefore, the cube will float with 1/4 L above the surface, not below or above the surface.
e) This option is not applicable as it is not a valid statement.
So, the correct answer is c) float with 1/4 L above the surface.
To determine whether the cube will sink or float in water, we need to compare the densities of the cube and the water.
Density is calculated by dividing the mass of an object by its volume.
Given that the cube is made of a substance that is 1/4 as dense as water, we can calculate its density relative to water.
Let's assume the density of water is represented by "Dw" and the density of the cube is represented by "Dc."
We know that Dw = 1 g/cm³ since water has a density of 1 g/cm³.
And we are told that Dc = 1/4 Dw.
Since Dc = (1/4)Dw, we can substitute Dw with 1 g/cm³:
Dc = (1/4)(1 g/cm³) = 1/4 g/cm³.
Now, let's compare the density of the cube to the density of water:
If the density of the cube (1/4 g/cm³) is less than the density of water (1 g/cm³), then the cube will float.
If the density of the cube is greater than the density of water, then the cube will sink.
Since 1/4 g/cm³ is less than 1 g/cm³, we can conclude that the cube will float.
However, the question asks how much of the cube will be above the water's surface.
The volume of the cube can be calculated as L³, where L is the length of the side of the cube.
Since L³ is the volume of the cube, it also represents the mass of the cube.
Using the density formula, Density = Mass/Volume, we can rearrange it to find the mass:
Mass = Density * Volume.
Let's substitute the values we have:
Density = 1/4 g/cm³ (Density of the cube)
Volume = L³ (Volume of the cube)
Mass = (1/4 g/cm³) * L³.
Since the cube is floating, the buoyant force acting on it must be equal to the weight of the cube.
The buoyant force can be calculated using Archimedes' principle:
Buoyant Force = Volume of Fluid Displaced * Density of Fluid * Acceleration Due to Gravity.
In this case, the fluid is water, and its density (Dw) is 1 g/cm³.
So, the Buoyant Force = L³ * (1 g/cm³) * 9.8 m/s².
For the cube to remain in equilibrium (not sinking or fully floating), the weight of the cube (Mass * Acceleration Due to Gravity) should be equal to the buoyant force.
Weight = Mass * Acceleration Due to Gravity.
Weight = (1/4 g/cm³ * L³) * 9.8 m/s².
From here, we can observe that the L³ term cancels out from both equations, leaving us with:
1/4 * 9.8 = 1 * 9.8 * (Volume of Fluid Displaced/Volume).
Simplifying further:
1/4 = Volume of Fluid Displaced/Volume.
Since the volume of the cube is (L³), we can rewrite the equation as:
1/4 = Volume of Fluid Displaced / (L³).
Now let's interpret the options given regarding the amount of the cube above the water's surface:
a) float with 1/2 L above the surface: This is not possible according to our calculations. Therefore, option a) is incorrect.
b) sink to the bottom: This is also not possible because we determined that the cube will float. Therefore, option b) is incorrect.
c) float with 1/4 L above the surface: This is correct based on our calculations. Since the cube is floating, it will displace a volume of water equal to its own volume. The remaining 3/4 L of the cube will be submerged underwater, leaving 1/4 L above the surface.
d) float with 1/4 L below the surface: This is incorrect since the cube will float above the surface. Therefore, option d) is incorrect.
e) float with 3/4 L: This is also incorrect. The cube will only have 1/4 L above the surface and the remaining 3/4 L will be submerged. Therefore, option e) is incorrect.
So, the correct answer is c) float with 1/4 L above the surface.