# Exploring the Relationship Between Buoyancy and Apparent Mass

• TkoT
In summary: Try using 1000 kg/m3 for the density of water. Also, there are typos in the numbers you posted. The correct numbers are in your answer. I can't follow your work. You don't seem to be following the pattern I have explained.
TkoT
Homework Statement
A cylindrical beaker of mass mb = 1.3 kg contains 1.5 x 10^3 ml of water. The beaker is placed on a scale and then a rock of mass mr = 2.2 kg, suspended by a massless string, is totally immersed in the water. The water level rises by 1.5 cm. The diameter of the beaker is 0.2m
a) What mass does the scale measure before the rock is lowered into the water?
b) What mass does the scale measure after the rock is lowered into the water?
Relevant Equations
F=mg
my solution:
a)

F(upward)=Fb +Fw
=(1.3+1.5)X9.8
=27.44N
total Mass = 2.8kg
b)

Volume increased = π(0.2/2)^2 x 1.5/100
=4.7x10-4 m^3

T+Fb =mg
T=mg-Fb
T=2.2x9.8 -1000 x 4.7x10-4 x 9.8
T=17.4N
T is the apparent weight of the rock, so the mass of the rock in the water is 1.74kg
So, the total mass measured by the scale = 1.74 +2.8 = 4.54kg

Question:
I am confused about the part b after I checked the answer. For me, I think the tension represents the apparent weight of the rock. So apparent mass of the rock can be obtained by the tension. But, In the answer, buoyant force is considered as the extra weight added to the scale. That confuses me and I don’t understand why.

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TkoT said:
Homework Statement:: A cylindrical beaker of mass mb = 1.3 kg contains 1.5 x 10^3 ml of water. The beaker is placed on a scale and then a rock of mass mr = 2.2 kg, suspended by a massless string, is totally immersed in the water. The water level rises by 1.5 cm. The diameter of the beaker is 0.2m
a) What mass does the scale measure before the rock is lowered into the water?
b) What mass does the scale measure after the rock is lowered into the water?
Relevant Equations:: F=mg

That confuses me and I don’t understand why.
Without the rock the scale reads just the weight of the water, ##m_{\text{water}}g##. When the rock is placed under water, the water exerts buoyant force BF up. By Newton's 3rd law, the rock exerts force BF down on the water. The scale reads the sum of the two because it must exert normal force ##N=BF+m_{\text{water}}g## up to keep the water from accelerating.

TkoT
You seem to be using 9.8m/s2 for g in some places and 10m/s2 in others.

## What is the relationship between buoyancy and apparent mass?

Buoyancy is the upward force exerted by a fluid on an object submerged in it, which counteracts the weight of the object. Apparent mass is the perceived weight of the object when submerged. The relationship is such that the apparent mass is equal to the actual mass of the object minus the mass of the fluid displaced by the object. This is why objects appear lighter in water.

## How do you calculate the buoyant force acting on an object?

The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. Mathematically, it is given by the formula: Buoyant Force = Density of Fluid × Volume of Displaced Fluid × Gravitational Acceleration.

## What factors affect the apparent mass of an object in a fluid?

The apparent mass of an object in a fluid is affected by the density of the fluid, the volume of the object, and the density of the object itself. The more dense the fluid, the greater the buoyant force, and thus the lower the apparent mass of the object. Conversely, the denser the object, the higher its apparent mass in the fluid.

## Why do objects feel lighter in water than in air?

Objects feel lighter in water than in air because water provides a greater buoyant force than air. This is due to the higher density of water compared to air. The buoyant force reduces the net weight of the object, making it feel lighter when submerged in water.

## Can the apparent mass of an object be zero? If so, under what conditions?

Yes, the apparent mass of an object can be zero if the buoyant force equals the actual weight of the object. This occurs when the weight of the fluid displaced by the object is equal to the weight of the object itself. In such a case, the object will be neutrally buoyant and will neither sink nor float.

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