Problem 73 - University Physics with Modern Physics

Question

A block of balsa wood placed in one scale pan of an equalarm balance is exactly balanced by a 0.115-kg brass mass in the other scale pan. Find the true mass of the balsa wood if its density is 150 \(\mathrm{kg} / \mathrm{m}^{3} .\) Explain why it is accurate to ignore the buoyancy in air of the brass but not the buoyancy in air of the balsa wood.

Step-by-Step Solution

Answer

True mass of balsa wood accounts for buoyancy; it's affected significantly unlike brass due to its low density.

Step 1: Understand the Problem

We need to find the true mass of the balsa wood when it's balanced by a 0.115 kg brass mass in an equal-arm balance. Given that the density of the balsa wood is 150 kg/m^3.

Step 2: Consider Buoyancy Effects

Since the problem involves air, we need to account for the buoyant force exerted by air. The balsa wood will experience a significant buoyant force due to its relatively low density compared to air, whereas the brass will experience negligible buoyancy due to its much higher density. Therefore, it's important to account for buoyancy in the balsa wood but not in the brass.

Step 3: Calculate the Volume of Balsa Wood

Use the formula for mass and density: \( \text{mass} = \text{density} \times \text{volume} \). Initially, both the balsa wood and brass have the same apparent mass of 0.115 kg, so: \( 0.115 = 150 \times V \), where \( V \) is the volume of the balsa wood. Solve for \( V \).

Step 4: Calculate the True Mass of Balsa Wood

The apparent mass includes the effect of buoyancy. True mass \( m_t = \text{apparent mass} + \text{buoyancy force} \). Calculate the buoyancy force as \( \text{density of air} \times g \times V \) where air density is approximately 1.2 kg/m^3. Use this to adjust the apparent mass of the balsa wood to its true mass.

Key Concepts

Buoyancy

Buoyancy is a fascinating physical phenomenon. It is the force exerted by a fluid on an object that is placed in it.
Imagine a sponge floating on water; it doesn't sink completely because the water pushes it upwards.
This push is what we call buoyancy. It arises because the pressure exerted by the fluid (in this case, air) increases with depth.
The bottom part of the object experiences more pressure than the top part, creating an upward force.

• In water, buoyancy is significant because water is denser than air.
• In our problem, we deal with buoyancy in air, which is usually less noticeable but becomes relevant for very light materials like balsa wood.

Understanding buoyancy helps when calculating how materials behave when submerged or simply placed in less dense fluids like air.

Density

Density is a crucial concept that connects mass and volume. It tells us how much mass is packed in a given space.
The formula for density is: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). This implies that density is measured in units like kg/m³.
In our scenario, the density of balsa wood is given as 150 kg/m³.

• This low density means it is much lighter than denser objects like brass which makes air buoyancy significant for balsa wood.
• Materials with high density, such as metals, pack a lot of mass into small volumes, reducing air buoyancy effects.

It's the contrast in density between balsa wood and air that makes buoyancy a factor worth considering when determining true mass.

Mass Calculation

Mass calculation involves a few steps and formulas. Initially, we consider mass as determined by the density and volume.
From density \( \rho \) (150 kg/m³ for balsa wood) and a known mass (of brass in this case 0.115 kg), you can find the volume:
\( \text{Mass} = \rho \times \text{Volume} \). Solving for volume gives us the space that balsa wood occupies.

• This is important because volume is needed to compute the buoyancy effect.
• The apparent mass of balsa in an equal-arm balance is initially observed as equal to the brass's mass, even though its true mass might differ due to buoyancy.

Mass calculation essentially enables understanding of the relationship between weight observed and the true essence of the object's mass influenced by factors like buoyancy.

Air Buoyancy Effects

Air buoyancy effects become notable when dealing with objects of significantly low density.
Air has a density of about 1.2 kg/m³. This means any object less dense than air would "float" more.
In practice, when considering air buoyancy, the force is computed as \( \text{density of air} \times \text{gravitational acceleration} \times \text{volume of the object} \).

• For balsa wood, this buoyant force is enough to alter its apparent mass observed in the balance.
• The effect on the brass is negligible due to its high density, so its apparent and true masses are nearly the same.

Understanding air buoyancy effects helps in adjusting the calculated mass to reflect the true mass by factoring in this seemingly invisible force.