Density and floating in equilibrium – problems and solutions

1. A block placed into two liquids with different types. In liquids A, 0.6 part of an object is in the liquid. In liquids B, 0.5 part of an object is in the liquid. Determine the ratio of the density of liquid A to liquid B.

__Known :__

Density of block = x

Part of the object in liquid A = 0.6

The density of liquid A = y

Part of object in liquid B = 0.5

The density of liquid B = z

__Wanted:__ The ratio of the density of liquid A to liquid B (y: z)

__Solution :__

Substitute x in equation 2 with x in equation 1 :

2. An object placed in liquid P, 0.5 part of the object is in liquid. When the object placed in liquid Q, the object is in liquid, as shown in the figure below. What is the ratio of the density of liquid P to liquid Q?

__Known :__

Density of object = x

Part of object in liquid P = 1/2 = 0.5

The density of liquid P = y

Part of the object in liquid Q = 1

The density of liquid Q = z

__Wanted:__ The ratio of the density of liquid P to liquid Q (y: z)

__Solution :__

Substitute x in equation 2 with x in equation 1 :

3. A wood floating in oil (density of oil is 800 kg/m^{3}). 4/5 parts of the wood are in the oil. What is the density of the wood?

__Known :__

Density of oil (ρ) = 800 kg/m^{-3}

Part of wood in oil = 4/5

__Wanted:__ Density of wood

__Solution :__

The density of wood is 640 kg/m^{3}.

4. The weight of a block is 50 N. In water, the weight of the block is 30 N. If the density of water is 10^{3} kg/m^{-3} what is the density of block.

__Known :__

Weight of block in air (w) = 50 N

Weight of block in water (w_{water}) = 30 N

Density of water (ρ_{water}) = 10^{3} kg/m^{3 }= 1000 kg/m^{3}

Acceleration due to gravity (g) = 10 m/s^{2}

Mass of block (m) = w/g = 50/10 = 5 kg

__Wanted :__ Density of block (ρ_{block})

__Solution :__

Buoyant force :

F_{A} = w – w_{water} = 50 – 30 = 20 N

Volume of block in water :

F_{A} = ρ g V

20 = (1000)(10) V

20 = (10,000) V

V = 20/10,000

V = 0.002 m^{3}

Density of block :

ρ = m/V = 5 / 0.002 = 2500 kg/m^{3 }

5. A wood block dipped in water. 25% of its part is above the surface of water and 75% is in water. Density of water is 1 g/cm^{3} and acceleration due to gravity is 10 m/s^{2}. Determine the density of the blocks.

A. 0.025 kg/m^{3}

B. 0.075 g/cm^{3}

C. 0.250 kg/m^{3}

D. 0.750 g/cm^{3}

__Known :__

Part of block above the surface of water = 25% = 25/100 = 0.25

The part of block in water = 75% = 75/100 = 0.75

The density of water (ρ_{water}) = 1 g/cm^{3} = 1000 kg/m^{3}

Acceleration due to gravity = 10 m/s^{2}

__Wanted :__ Density of block (ρ_{block})

__Solution :__

Volume of block in water = the ratio of the density of block to the density of water

The correct answer is D.

**What is density, and how is it defined mathematically?****Answer:**Density is a measure of how much mass is contained in a given volume. Mathematically, it’s defined as $mass/volume $.

**How does the density of an object determine if it will float or sink in a fluid?****Answer:**If the density of an object is less than the density of the fluid in which it is placed, the object will float. Conversely, if the object’s density is greater than the fluid’s density, it will sink.

**What is the principle behind the floating or sinking of an object in a fluid?****Answer:**The principle is Archimedes’ Principle, which states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces.

**If an ice cube floats in water with a portion of it above the water level, what does this indicate about the relative densities of ice and water?****Answer:**This indicates that the density of ice is less than the density of water. That’s why the ice cube floats, and only a portion of it remains submerged to displace an amount of water equal in weight to the ice cube.

**Why do heavy steel ships float on water while a small steel nail might sink?****Answer:**The overall density of a steel ship, including its hollow portions filled with air, is less than the density of water. In contrast, the density of a solid steel nail is greater than the density of water, causing it to sink.

**How is the concept of “neutral buoyancy” different from floating or sinking?****Answer:**Neutral buoyancy occurs when an object’s weight is exactly balanced by the buoyant force, causing the object to remain suspended within the fluid without rising or sinking.

**Why does a helium balloon float in the air?****Answer:**The density of helium is significantly less than the density of air. As a result, the weight of the helium balloon (including the balloon material and the helium inside) is less than the weight of the air it displaces, making it float.

**What happens to the density of an object when it’s compressed into a smaller volume while its mass remains constant?****Answer:**The density will increase since density is defined as mass divided by volume. If the volume decreases and the mass remains constant, the ratio (density) will increase.

**How does salinity affect the density of water?****Answer:**Increasing the salinity of water (i.e., adding more salt) will increase its density. This is why objects tend to float more easily in saltwater (like the Dead Sea) compared to freshwater.

**If an object is floating in equilibrium in a fluid, how do the buoyant force and the gravitational force on the object compare?**

**Answer:**When an object is floating in equilibrium, the buoyant force (upward force due to the fluid) is equal in magnitude and opposite in direction to the gravitational force (weight) acting on the object.