Test: Archimedes Principle - 1 - EmSAT Achieve MCQ

Concept:

When the density of the body is less than that of the water, it floats. According to Archimedes' principle the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.

For floating bodies, the weight of the body = buoyancy force

Weight of the body = Immersed volume x Unit weight of water.

Calculations:

Given, 

L = 2m, B = 1 m, h = 1 m, γ = 10 kN/m³, d = 0.5 m

Weight of body = Buoyancy Force

Buoyancy (B) = Immersed volume × Unit weight of water

W = L x B x d × γ 

W = 2 x 1 x 0.5 × 10 = 10 kN 

Concept:

Pressure intensity is given as, P = ρgh

The height of the imaginary free surface of the water above the bottom of the gate equivalent to a pressure intensity of 196.2 kN/m2 is, 

The total pressure on gate is, P = wAx̅ 

Where, A = area of gate, x̅ = vertical distance of CG from free surface 

Vertical distance of CP from free surface is,

Where, I_G = MOI at the center of axis, 

Calculation: 

Given:

ρ = 1000 kg/m³, g = 9.81 m/s², A = 4 × 4, θ = 90°, P = 196.2 kN/m². 

The height of the imaginary free surface of the water,

The total pressure on gate is, P = wAx̅ 

Vertical distance of CG from free surface, x̅ = 20 - 2 = 18 m

Vertical distance of CP from free surface is, 

The location of center of pressure below the centroid of gate is, 

y = h - x̅ 

Concept:​

• A floating object displaces an amount of water equal to its own weight. 
• Since water expands when it freezes, one unit (of weight) of frozen water has a larger volume than one unit of liquid water.

Explanation:

• As the density of oil water and ice is as follows:

d_ice = 0.91 g/cm³, d water = 1 g/cm³, d_oil is = 0.85 g/cm³

• If the ice is dropped in oil, as the density of ice is more it will sink in the oil in the container.
• ​Now when the ice melts it gets converted into water, but as the water density is also more than oil, so the water will also sink and as a result, the level of oil will rise.
  So here, overall level of the oil will rise.
  Note: According to the official answer key option B is correct.

CONCEPT:

• Archimedes' principle: A body at rest in a fluid is acted upon by a force pushing it upward called the buoyant force, which is equal to the weight of the fluid that the body displaces.

​The buoyant force (FB) is given by,

F_B = ρ_fluid × g × V_object

​EXPLANATION:

The floating ice will displace some volume of the water.

From Archimedes' principle, F_B = ρ_water × g × V_ice      ----(1)

Weight of the displaced fluid (water) = mg = ρ_water × V_{dispalced water} × g      ----(2)

For an object to stay afloat, the forces acting on it must be balanced ⇒ (1) = (2)

ρ_water × g × V_ice = ρ_water × V_water × g 

⇒  V_ice = V_{dispalced water}

• Therefore, when the ice melts, there will be no change in the water level as the melted ice will occupy the same volume as that of displaced water.

Concept:

When a body is either wholly or partially immersed in a fluid, a lift is generated due to the net vertical component of hydrostatic pressure forces experienced by the body. This lift is called the buoyant force and the phenomenon is called buoyancy.

The Archimedes principle states that the buoyant force on a submerged body is equal to the weight of the liquid displaced by the body and acts vertically upward through the centroid of the displaced volume.

Thus, the net weight of the submerged body, (the net vertical downward force experienced by it) is reduced from its actual weight by an amount that equals the buoyant force.

F_B = ρghA = ρgV

Weight of cube = buoyancy force

ρ_iceV_iceg = ρ_wV_VDg

Calculation:

Given:

ρ_ice = 920 kg/m³, ρw =1000 kg/m³

Let the x be the volume of iceberg floats above the water surface.

ρ_iceVg = ρ_wg(V - x)

920 × V × g = 1000 × g × (V - x)

0.92V = (V - x)

x = 8%

Concept:

Archimedes Principle: When a body is fully or partially dipped in fluid, the fluid exerts a contact force on the body. The resultant of this contact force is known as buoyancy or buoyant force.

• This force acts vertically upwards (opposite to the weight of the body).
• Mathematically, F = Vρg [where, V = volume of liquid displaced, ρ = density of the liquid, g = acceleration due to gravity.]
• A body floats on a fluid if the average density of the body is less than that of fluid.
• The weight of the liquid displaced by the immersed part of the body must be equal to the weight of the body.

Calculation:

Given:

m = 700 g = 0.7 kg, a = 10 cm 

In floating bodies,

Weight of fluid = Buoyant force

Buoyant force = weight of water displaced. If V is the volume of cube inside the water, then the weight of water displaced = Vρg.

⇒ Vρg = 0.7 g

Total volume of the cube = (10 cm) ³ = 1000 cm³

∴ Volume outside the water = 1000 – 700 = 300 cm³

CONCEPT:

• Archimedes Principle: The upward buoyant force that is exerted on a body immersed in a fluid, whether partially or fully submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the centre of mass of the displaced fluid.
  • The value of thrust force is given by the Archimedes law which was discovered by Archimedes of Syracuse of Greece. When an object is partially or fully immersed in a liquid displaced by it.
  • Archimedes' principle tells us that this loss of weight is equal to the weight of liquid the object displaces. If the object has a volume of V, then it displaces a volume V of the liquid when it is fully submerged. If only a part of the volume is submerged, the object can only displace that much liquid.
• Archimedes Principle Formula: In simple form, the Archimedes law states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. Mathematically written as:

F_b =  ρ × g × V

Where F_b is the buoyant force, ρ is the density of the fluid, V is the submerged volume, and g is the acceleration due to gravity

​EXPLANATION:

• When a body partially and fully emerged in any fluid it experiences an upward force that is equal to the weight of the fluid displaced by it and it is called buoyant force.
• Hence option 4 is the correct answer.

CONCEPT:

• A simple device that uses hot air or light gas to move up in the air is called a balloon.
• Archimedes principle: Any object which is floating in a fluid is acted upon by an upward force whose magnitude is equal to the weight of the fluid displaced.
  • The balloon is based on the principle of Archimedes.

EXPLANATION:

• When the air is being heated the density of that air gets reduced compared to the density of surrounding air or we can use the gas having less density than that of air.
• Due to the difference in density, the balloon lifted above the ground and moves up in the air.
• Thus a balloon is based on the Archimedes principle. So option 2 is correct.

EXTRA POINTS:

• Bernoulli's principle: For a streamlined flow of an ideal liquid in a varying cross-section tube the total energy per unit volume remains constant throughout the fluid.
• Pascal’s principle: Pascal’s Law is the principle of transmission of fluid-pressure.
  • It says that "a pressure exerted anywhere in a point of the confined fluid is transmitted equally in all directions throughout the fluid”.

The correct answer is The upthrust on both balls will be the same.

Key Points

• The balls of iron and aluminium of the same diameter will have the same volume.
• So, the buoyant force exerted by water on both the balls will be the same.
• Buoyant Force is defined as the upward force exerted by a fluid on the immersed body. 
• It is also called upthrust.

Additional Information

• Archimedes’ principle states that when an object immersed in fluid experiences a buoyant force that is equal in magnitude to the force of gravity on the displaced fluid.
• When an object is immersed wholly or partially in a fluid, then there is some apparent loss in its weight. This loss in weight is equal to the weight of the liquid displaced by the body.  

• If the solid with a density of 900 kg/m³ is able to float in the oil and the oil, in turn, floats on water with a density of 1000 kg/m³, then indeed the density of the oil must be between the densities of the solid and the water.
• A solid of density 900 kg/m3 floats in oil as the density of solid is less than that of oil.
• And the oil floats in water as its density is less than that of water.
• So, the density of oil must be between 900 kg/m3 to1000 kg/m3.
• Therefore, the density of oil could be 950 kg/m3.
• This density would allow the solid to float in the oil, and the oil would float on the water.

Thus, A solid of density 900 kg/m3 floats in oil. The oil floats on water of density 1000 kg/m3. The density of oil in kg/m3 could be 950 kg/m3.