Problem 24 - Applied Physics

Question

If racing alcohol has a mass density of \(790 \mathrm{~kg} / \mathrm{m}^{3}\), what mass will a 1250 -litre tank hold?

Step-by-Step Solution

Answer

The tank will hold 987.5 kg of racing alcohol.

Step 1: Understanding Mass Density

Mass density refers to the mass per unit volume of a substance. It is given by the formula \( \text{density} = \frac{\text{mass}}{\text{volume}} \). In this exercise, the mass density of racing alcohol is given as \(790 \text{ kg/m}^3\).

Step 2: Converting Volume to Cubic Meters

Since the mass density is given in kilograms per cubic meter (\(\text{kg/m}^3\)), we need to convert the volume from liters to cubic meters. \(1\) liter is equivalent to \(0.001\) cubic meters. Therefore, \(1250\) liters is \(1250 \times 0.001 = 1.25\) cubic meters.

Step 3: Calculating the Mass

Using the formula \( \text{mass} = \text{density} \times \text{volume} \), we can calculate the mass of the alcohol in the tank. Substituting the known values, mass \(= 790 \text{ kg/m}^3 \times 1.25 \text{ m}^3 = 987.5 \text{ kg}\).

Key Concepts

Volume Conversion

Volume conversion is an essential step when solving problems involving mass density and volume measurements, especially when units don't initially align. In physics, it's common to encounter situations where you need to convert between different units of volume.
For example, in this exercise, we start with a volume given in liters. However, the mass density is provided in cubic meters, specifically in kg/m³. To make the calculations straightforward, we convert the initial volume from liters to cubic meters.
The conversion factor between liters and cubic meters is simple:

• 1 liter = 0.001 cubic meters

Therefore, if you have a volume of 1250 liters, you multiply by 0.001 to convert it to cubic meters, resulting in 1.25 cubic meters. Understanding this conversion is crucial for using the mass density formula effectively.

Mass Calculation

Mass calculation involves using the relationship between mass, density, and volume, which is elegantly summarized in the formula:
\[ \text{mass} = \text{density} \times \text{volume} \]This straightforward equation allows you to calculate the mass of any substance if you know its density and volume. In our problem, the mass density of racing alcohol is given as 790 kg/m³.
With our converted volume of 1.25 cubic meters, we can now find the mass by plugging the values into the formula:

• Mass = 790 kg/m³ × 1.25 m³

This calculation gives a mass of 987.5 kg. The process is simple, but understanding this formula is foundational for tackling density-related problems.

Physics Problem Solving

Physics problem solving often involves breaking down a problem into manageable steps. Grasping the underlying principles and applying them in sequence can make complex problems much simpler.
In our example exercise:

• We start by understanding what mass density means and how it relates to the other quantities we've been given.
• Next, we ensure our units are consistent, converting volume from liters to cubic meters so that it matches the unit used in our density measurement.
• Finally, we use the appropriate formula to compute the desired quantity—in this case, mass.

Following these steps systematically is a vital approach in physics, ensuring you address all aspects of the problem correctly and efficiently. This structure not only leads to the correct answer but also builds a deeper understanding of the concepts involved, preparing you for more complex problems in the future.