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Chapter 5: Problem 14

What is the mass in kilograms of \(1 \mathrm{~L}\) of water? What is its weight in newtons?

Short Answer

Expert verified
The mass of 1 L of water is 1 kg, and its weight is 9.81 newtons.

Step by step solution

01

Determining the Mass of Water

The density of water is approximately \(1 \text{g/cm}^3\). Since there are \(1000 \text{cm}^3\) in \(1 \text{L}\), the mass of \(1 \text{L}\) of water can be calculated by multiplying the density of water by the volume in cubic centimeters. Therefore, mass of water in grams is given by \(1 \text{g/cm}^3 \times 1000 \text{cm}^3 = 1000 \text{g}\). To convert grams to kilograms, divide by 1000. Thus, \(1000 \text{g} \times \frac{1 \text{kg}}{1000 \text{g}} = 1 \text{kg}\).
02

Calculating the Weight in Newtons

The weight of an object on Earth is the force due to gravity acting on its mass. This force is calculated by multiplying the mass (\(m\)) by the acceleration due to gravity (\(g\)), which is approximately \(9.81 \text{m/s}^2\). Since we've determined that the mass is \(1 \text{kg}\), the weight \(W\) in newtons is given by \(W = m \times g = 1 \text{kg} \times 9.81 \text{m/s}^2 = 9.81 \text{N}\).

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Key Concepts

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

Mass-Volume Relationship
Understanding the mass-volume relationship is crucial when working with substances in different physical states. For water, a liquid, the density is typically used to express this relationship. Density is defined as mass per unit volume, and for water, it's approximately 1 g/cm3.

Here's a practical example: to find the mass of water, you multiply its volume by the density. Since one liter of water is equivalent to 1000 cm3, and water's density is 1 g/cm3, you easily see that one liter of water has a mass of 1000 grams. These essential concepts allow us to relate mass to volume in a tangible and quantifiable way, laying the foundation for more complex scientific and engineering calculations.
Unit Conversion
Unit conversion is a fundamental skill in science and everyday life, enabling us to switch between different measurement systems. When working with mass and volume, it's often necessary to convert units for easier computation and understanding.

In the example provided, we convert the mass of water from grams to kilograms. Since 1000 g = 1 kg, we divide the mass in grams by 1000 to get the mass in kilograms. This skill is indispensable because scientists and engineers frequently need to work within different metric and imperial units, and being adept at conversion makes problem-solving much more manageable.
Calculating Weight
Calculating weight involves understanding that it is the result of gravity acting on mass. To calculate the weight on Earth, we use the formula W = m 脳 g, where W is weight in newtons, m is mass in kilograms, and g is the acceleration due to gravity.

Take our example of 1 kg of water; its weight can be easily calculated by multiplying by the standard Earth gravity, 9.81 m/s2. Therefore, the water's weight is 9.81 N. This calculation is essential in fields like engineering, where knowing the weight of materials is crucial to design.
Acceleration Due to Gravity
The acceleration due to gravity is a constant force that acts on all objects near the Earth's surface, drawing them toward the center of the planet. This acceleration is approximately 9.81 m/s2 and is a critical factor when calculating the weight of an object.

It's important to note that while mass is a measure of the amount of matter in an object and remains unchanged regardless of location, weight can vary depending on the strength of gravity. Hence, weight calculated on Earth would differ from that on the Moon or another planet, whereas the mass would remain the same.

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