1.2: Fundamental Units and Standard Units
Vocabulary
- dimensional analysis: Checking your mathematical equations by keeping units with every number, or predicting units based on the relation of other units.
- length: The measurement of the extent of something along its greatest dimension.
- mass: A property of matter equal to the measure of an object's resistance to changes in either the speed or direction of its motion. The mass of an object is not dependent on gravity and therefore is different from but proportional to its weight.
- meter stick: A ruler exactly one meter in length.
- scalar: A quantity, such as mass, length, or speed that is completely specified by a single number. It is said to have magnitude, but no direction.
- SI units: The most modern version of the metric system, also known as the International System of Units (SI).
- time: An ordered continuum in which events occur in succession from the past through the present to the future.
- vector: A quantity, such as velocity or force, that must be completely specified by both a magnitude and a direction. For example, if an object is moving, we must know both the amount of motion as well as the direction of motion to know the vector quantity velocity.
Equations Including Units
Mathematics is central to science in general and especially physics. The math in physics is always connected to real physical quantities, however. In a math class, an equation might read:
\begin{align*}24 - \frac{1}{2} \times 10 \times (2)^2 = 10\end{align*}
However, in physics all numbers represent a specific physical quantity and they have units.
\begin{align*}24m - \frac{1}{2} \times 10 \frac{m}{s^2} \times (2s)^2 = 4m\end{align*}
Each of the quantities in the equation has abbreviations for units beside it. For example, this is the formula for how high an object would be after dropping for two seconds from the top of a six-story building.
Standard Units—SI Units (Standard International Units)
Especially in scientific work, it is vital that we use a consistent and logical set of units. For almost all scientists, that system is the International System of Units, abbreviated SI. This the most modern form of the metric system. It is used universally for all scientific work. It is also used on a daily basis in almost all the world. The three countries that do not officially use metric units in some form today are Burma, Liberia, and the United States.
Map of countries not officially using the metric system worldwide.
Students often learn the basics of metric units, but are not fluent in them. Especially in physics, it is important not just to perform abstract calculations, but to be able to mentally picture what measurements look like.
- How heavy is a 10kg weight?
- How short is someone who is 155cm tall?
The standard unit of length used in physics today is the meter, which is roughly the distance from the tip of your nose to the end of your outstretched arm. The meter sticks you have in your classroom do not have this accuracy.
The standard unit of mass is the kilogram. A one-kilogram mass corresponds to a weight of approximately 2.2 pounds. Mass and weight are not the same property. Weight is the product of the gravitational acceleration, \begin{align*}g\end{align*}, and the mass of the object: \begin{align*}W = mg\end{align*}. The corresponding standard weight unit is the Newton, where 4.45 Newtons is equal to 1 pounds.
The standard unit of time is the second, which is familiar to most students.
The standard unit of temperature is the degree Celsius. One degree Celsius is the same as one degree Kelvin, but they measure from a different starting point. Celsius measures up from the freezing point of water, while Kelvin measures up from the point called "absolute zero".
These fundamental units are combined into derived units, such as \begin{align*}\frac{\text{length}}{\text{time}}\end{align*} for speed. Speed and velocity have the derived units of \begin{align*}\frac{\text{length}}{\text{time}}\end{align*}. Other derived units include force, energy, voltage, and so forth.
Vectors and Scalars
If we wish to fully define the motion of an object we must state the object’s magnitude and the object’s direction. In the case of motion, the magnitude is the speed of the object. Quantities such as displacement are represented by net distance (magnitude) and direction. Any quantity requiring a magnitude and a direction is called a vector quantity. Force is another vector quantity. For example: A 150 N punch (magnitude) to the stomach (direction, loosely speaking!). Quantities that require only a magnitude for their complete description are called scalars. Mass and temperature are examples of scalar quantities.
A vector requires both a magnitude and direction.
Dimensional Analysis
Dimensional analysis is the method of using fundamental or standard units in detecting an error and discerning the correct relationship between physical quantities. As a simple example, recall our old friend \begin{align*}d = rt\end{align*}, where \begin{align*}d\end{align*} is a distance, \begin{align*}r\end{align*} is a rate, and \begin{align*}t\end{align*} is time. If we accidentally rewrite the equation as \begin{align*}r = t/d\end{align*}, we see that the units are inconsistent since m/s do not equal s/m. If the derived units are not the same on each side of the equation, we know that a mistake was made. The converse is not true. If the units agree, a mistake may still have been made. (Why?)
Check Your Understanding
1. The density of mercury is 13.5 g/cm^{3}. What is the formula for density?
Answer: The units of density are given as g/cm^{3}. Grams are a measure of mass, and cm^{3} is a measure of volume. Density therefore has units of mass/volume. So the formula for density is: \begin{align*}d = m/V\end{align*}, where \begin{align*}d\end{align*} is density. Of course, there may be a “dimensionless” quantity in this formula. Not all formulas have only physical quantites in them. Consider the volume, \begin{align*}V\end{align*}, of a sphere: \begin{align*}V = \frac{4}{3} \pi r^3\end{align*}. Here, \begin{align*}r\end{align*} is the radius of the sphere and it has units of length. But the quantity \begin{align*}\frac{4}{3} \pi\end{align*} has no units. It is a “pure number.” Dimensional analysis cannot determine what, if any, pure numbers are in a formula. But, at the very least, a statement of proportionality is always possible, as in the case of \begin{align*}V\end{align*} is proportional to \begin{align*}r^3\end{align*}.
2. In the formula, \begin{align*}x = \frac{1}{2} at^2\end{align*}, \begin{align*}x\end{align*} has units of meters and \begin{align*}t\end{align*} has units of seconds. What are the units of \begin{align*}a\end{align*}?
Answer: The acceleration, \begin{align*}a\end{align*}, has units of m/s^{2}.
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Learning Objectives
- List and use fundamental units in the study of mechanics.
- List and use standard units in the study of mechanics.
- Use dimensional analysis.