Lesson 1.1: Quantities and SI Units in Mechanics
Introduction
In this lesson, we will establish the fundamental concepts of quantities, units, and modeling in mechanics. Understanding these concepts is crucial for tackling more complex problems later in the course. Our key learning objectives for this lesson include:
- Identifying the fundamental SI quantities and their corresponding units used in mechanics: length (metre), time (second), and mass (kilogram).
- Understanding derived quantities and their units, such as velocity, acceleration, force (newton), weight, and moment.
- Learning how to check that an equation is dimensionally consistent and that answers carry the correct units.
- Stating the SI unit for each fundamental and derived quantity encountered in mechanics.
- Converting between related units and expressing a quantity to an appropriate degree of accuracy.
Fundamental SI Quantities and Units
Length
Length is one of the fundamental quantities measured in mechanics. In the International System of Units (SI), the unit of length is the metre (m). The metre was historically defined based on the Earth’s circumference; however, it is now defined in terms of the speed of light.
$1 \text{ m}$ = 1/299,792,458 \text{ seconds}\, \text{(speed of light in vacuum)}
It is essential to grasp the significance of the metre as a base unit, as it serves as the basis for measuring longer or shorter distances, which are often expressed in multiples or fractions of a metre, such as kilometers (1 km = 1000 m) and millimetres (1 mm = 0.001 m).
Time
The fundamental unit of time in the SI system is the second (s). Like the metre, the second has a precise definition based on natural phenomena, specifically the vibrations of cesium atoms.
$1 \text{ s}$ = \text{duration of 9,192,631,770 cycles of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom}
Time is crucial for any mechanical system since it allows us to analyze motion, changes in position, and other dynamic properties.
Mass
Mass is a fundamental quantity that represents the amount of matter in an object. The SI unit for mass is the kilogram (kg). The kilogram was originally defined as the mass of a specific platinum-iridium cylinder stored in France. Currently, it is defined based on fixed numerical values of fundamental constants.
$1 \text{ kg}$ = \text{mass of the International Prototype of the Kilogram} $\approx 1000$ $\text{ g}$
Mass is directly related to force and Newton's laws of motion, as it is an essential component in calculating the dynamics of an object.
Derived Quantities and Units
Derived quantities are those that are not fundamental but are derived from the fundamental quantities through mathematical relationships. Below, we discuss some important derived quantities in mechanics and their respective units:
Velocity
Velocity is a vector quantity that denotes the rate of change of position with respect to time. It is given by the formula:
$v = \frac{\Delta x}{\Delta t}$
where $ \Delta x $ represents the change in position (in metres) and $ \Delta t $ represents the change in time (in seconds). This leads us to the unit of velocity:
$\text{Velocity} = \frac{\text{metres}}{\text{seconds}} = \text{m/s}$
Example: Evaluating Velocity
Consider a car that travels 100 meters in 5 seconds. We can calculate its average velocity as follows:
v = $\frac{100 \text{ m}}{5 \text{ s}}$ = $20 \text{ m/s}$
This means that the car travels at an average velocity of 20 metres per second.
Acceleration
Acceleration is another vector quantity that represents the rate of change of velocity with time. The formula for acceleration is given by:
$a = \frac{\Delta v}{\Delta t}$
where $ \Delta v $ is the change in velocity and $ \Delta t $ is the time interval. The unit of acceleration in SI is:
$\text{Acceleration} = \frac{\text{m/s}}{\text{s}} = \text{m/s}^2$
Example: Evaluating Acceleration
If a car's velocity changes from 0 to 20 m/s in 4 seconds, we can find the acceleration as follows:
a = $\frac{20 \text{ m/s} - 0 \text{ m/s}}{4 \text{ s}}$ = $5 \text{ m/s}^2$
Here, the car accelerates at $5 \text{ m/s}^2$.
Force
Force is a derived vector quantity defined as the interaction that causes an object to accelerate. According to Newton's second law, force can be expressed as:
$F = m \cdot a$
where $ F $ is the force (in newtons), $ m $ is the mass (in kilograms), and $ a $ is the acceleration (in m/s²). The unit of force is the newton (N), which can be defined as:
$1 \text{ N}$ = $1 \text{ kg}$ $\cdot 1$ $\text{ m/s}^2$
Example: Calculating Force
If a mass of 2 kg is accelerating at $3 \text{ m/s}^2$, the force exerted can be calculated as:
F = $2 \text{ kg}$ $\cdot 3$ $\text{ m/s}^2$ = $6 \text{ N}$
Thus, a force of 6 newtons is acting on the object.
Weight
Weight is the force exerted by gravity on an object and can be calculated by the formula:
$W = m \cdot g$
where $ W $ is the weight (in newtons), $ m $ is the mass (in kilograms), and $ g $ is the acceleration due to gravity, approximately $ 9.81 \text{ m/s}^2$ on the surface of the Earth.
Example: Calculating Weight
For a mass of 10 kg, the weight can be calculated as:
W = $10 \text{ kg}$ $\cdot 9$.$81 \text{ m/s}^2$ = $98.1 \text{ N}$
The weight of the object is approximately 98.1 newtons.
Moment
The moment about a point is a measure of the rotational effect of a force. The moment $ M $ can be calculated using the formula:
$M = F \cdot d$
where $ F $ is the force applied (in newtons) and $ d $ is the perpendicular distance from the line of action of the force to the pivot point (in metres). The unit of moment is the newton metre (N·m).
Example: Calculating Moment
If a force of 5 N is applied at a distance of 2 m from the pivot, the moment can be calculated as:
M = $5 \text{ N}$ $\cdot 2$ $\text{ m}$ = $10 \text{ N·m}$
Thus, the moment about the pivot is $ 10 \text{ N·m}$.
Dimensional Consistency
Dimensional consistency means that both sides of an equation must have the same dimensions, allowing for a comparison of physical quantities. For instance, in the equation of motion, the dimensions of displacement must equal the dimensions on both sides of the equation.
Example: Dimensional Analysis
In the equation:
$d = v \cdot t$
Let’s check for a simple dimensional analysis where displacement $ d $ is in metres and velocity $ v $ is in m/s while time $ t $ is in seconds.
- Dimensions of $ d $: [L] (length in metres)
- Dimensions of $ v \cdot t $: $$\text{[L]}$ = $\text{[L]/[T]}$ $\cdot$ $\text{[T]}$ = $\text{[L]}
Both sides are consistent and this equation holds true.
Conclusion
In this lesson, we reviewed key concepts related to quantities and units in mechanics. You learned about fundamental SI quantities such as length, time, and mass, as well as derived quantities like velocity, acceleration, force, weight, and moment. By understanding the importance of these quantities and practicing dimensional analysis, you can ensure accuracy in problem-solving and modeling physical systems effectively.
Study Notes
- Fundamental Si units: Length (m), Time (s), Mass (kg).
- Derived units: Velocity (m/s), Acceleration (m/s²), Force (N = kg·m/s²), Weight (N), Moment (N·m).
- Verify dimensional consistency in equations by ensuring both sides have the same dimensions.
- Conversions between units are often necessary; practice converting units as required.
- Express results to a suitable number of significant figures to reflect measurement accuracy.