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A 2.0-cm cube of metal is suspended by a fine thread attached to a scale. The cube appears to have a mass of \(47.3 \mathrm{~g}\) when measured submerged in water. What will its mass appear to be when submerged in glycerin, sp gr = 1.26? [Hint: Find \(\rho\) too.]

Short Answer

Expert verified
Apparent mass in glycerin is 45.22 g.

Step by step solution

01

Understanding the Problem

We need to find the apparent mass of a metal cube in different liquids using the principle of buoyancy. We know the apparent mass when submerged in water, and we need to find it when submerged in glycerin.
02

Calculate Volume of the Cube

The cube is 2.0 cm on each side. Calculate its volume using the formula for the volume of a cube: \[ V = a^3 = (2.0 ext{ cm})^3 = 8.0 ext{ cm}^3 \].
03

Calculate Buoyant Force in Water

The apparent mass of the cube is given in water as 47.3 g. The buoyant force can be calculated by subtracting the weight of the cube in water from its actual weight. Let's denote the actual mass of the cube as \( m \) g.
04

Finding the Actual Mass of the Cube

The buoyant force in water equals the weight of the displaced water volume, which is equal to the volume of the cube times the density of water (1g/cm³). Let the actual mass of the cube be \( M \).Thus, \( 8 ext{ g} = M - 47.3 ext{ g} \).Solving for \( M \), \( M = 47.3 + 8 = 55.3 ext{ g} \).
05

Calculate Density of Glycerin

Given that the specific gravity of glycerin is 1.26, its density \( \rho \) is \( 1.26 \times 1 ext{ g/cm}^3 = 1.26 ext{ g/cm}^3 \).
06

Calculate Buoyant Force in Glycerin

The buoyant force when the cube is submerged in glycerin is the weight of the volume of glycerin displaced.\[ F_{glycerin} = V \times \rho_{glycerin} = 8.0 ext{ cm}^3 \times 1.26 ext{ g/cm}^3 = 10.08 ext{ g} \].
07

Calculate Apparent Mass in Glycerin

The apparent mass of the object in glycerin would be the actual mass minus the buoyant force in glycerin:\[ M_{apparent} = 55.3 ext{ g} - 10.08 ext{ g} = 45.22 ext{ g} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Apparent Mass
The concept of apparent mass is centered around the idea of how an object's mass changes when it is submerged in a fluid. Apparent mass refers to the weight an object seems to have when it's lifted under water or any liquid other than air. This feels less than its actual mass because of the buoyant force, which counteracts some of the object's weight.
To calculate apparent mass, you subtract the buoyant force exerted by the liquid on the submerged object from the object's actual mass. It's important to note:
  • The buoyant force is equal to the weight of the fluid displaced by the object.
  • In our example, the cube's apparent mass in water was 47.3 g, and it became 45.22 g when submerged in glycerin due to differences in buoyancy.
This alteration in mass gives an illusion that the object weighs less, when in reality, it stays the same.
Specific Gravity
Specific gravity is a concept used to compare the density of a substance to a reference substance, usually water for liquids and solids. It helps us understand how dense a fluid is compared to another. If a liquid has a specific gravity greater than 1, it's denser than water.
In the exercise, glycerin has a specific gravity of 1.26. This means glycerin is 1.26 times denser than water. Specific gravity is calculated with the formula:
  • Specific Gravity = Density of Substance / Density of Water
Recognizing specific gravity helps to foresee how the buoyant force will change when an object is submerged in different fluids. A higher specific gravity indicates a stronger buoyant force. This was evident when the object switched from water to glycerin.
Density Calculation
Density is a fundamental concept that determines how much mass is contained in a given volume. It is expressed as mass per unit volume and is crucial when calculating buoyant forces and understanding specific gravity.
  • The density formula is: \[\text{Density} (\rho) = \frac{\text{Mass}}{\text{Volume}}\]
  • In the exercise, the density of glycerin was calculated using its given specific gravity: \[1.26 \text{ g/cm}^3 = 1.26 \times 1 \text{ g/cm}^3\]
Understanding density helps explain why objects float or sink, and assists in assessing the apparent mass in different fluids. It's a core concept in studying buoyancy and helps clarify how different materials interact with each other when placed in a fluid.

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Most popular questions from this chapter

A wooden cylinder has a mass \(m\) and a base area \(A\). It floats in water with its axis vertical. Show that the cylinder undergoes SHM if given a small vertical displacement. Find the frequency of its motion. When the cylinder is pushed down a distance \(y\), it displaces an additional volume Ay of water. Because this additional displaced volume has mass \(A y_{\rho w}\), an additional buoyant force \(A y_{\rho w g}\) acts on the cylinder, where \(\rho_{w}\) is the density of water. This is an unbalanced force on the cylinder and is a restoring force. In addition, the force is proportional to the displacement and so is a Hooke's Law force. Therefore, the cylinder will undergo SHM, as described in Chapter 11 . Comparing \(F_{B}=A \rho_{w} g y\) with Hooke's Law in the form \(F=k y\), we see that the elastic constant for the motion is \(k=A \rho_{w} g .\) This, acting on the cylinder of mass \(m\), causes it to have a vibrational frequency of $$ f=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}=\frac{1}{2 \pi} \sqrt{\frac{A \rho_{w} g}{m}} $$

A beaker contains oil of density \(0.80 \mathrm{~g} / \mathrm{cm}^{3}\). A \(1.6-\mathrm{cm}\) cube of aluminum \(\left(\rho=2.70 \mathrm{~g} / \mathrm{cm}^{3}\right)\) hanging vertically on a thread is submerged in the oil. Find the tension in the thread.

A barrel will rupture when the gauge pressure within it reaches 350 \(\mathrm{kPa}\). It is attached to the lower end of a vertical pipe, with the pipe and barrel filled with oil \(\left(\rho=890 \mathrm{~kg} / \mathrm{m}^{3}\right)\). How long can the pipe be if the barrel is not to rupture? From \(P=\rho g h\) we have $$ h=\frac{P}{\rho g}=\frac{350 \times 10^{3} \mathrm{~N} / \mathrm{m}^{2}}{\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)\left(890 \mathrm{~kg} / \mathrm{m}^{3}\right)}=40.1 \mathrm{~m} $$

A spring whose composition is not completely known might be either bronze (sp gr \(8.8\) ) or brass (sp gr \(8.4\) ). It has a mass of \(1.26\) g when measured in air and \(1.11\) g in water. Which is it made of?

A tank containing oil of sp gr \(=0.80\) rests on a scale and weighs \(78.6 \mathrm{~N}\). By means of a very fine wire, a \(6.0 \mathrm{~cm}\) cube of aluminum, sp gr \(=2.70\), is submerged in the oil. Find \((a)\) the tension in the wire and \((b)\) the scale reading if none of the oil overflows.

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