Definitions of Mechanical Quantities

Understanding mechanics requires precise definitions of the quantities involved. These fall into two categories: fundamental quantities that cannot be defined in terms of others, and derived quantities built from the fundamental ones.

The Three Undefinable Quantities

Three quantities in mechanics are undefinable in terms of any other quantity:

Mass (m)

• Unit: kilogram (kg)
• Meaning: The amount of material in an object
• Important distinction: Mass refers to amount of matter, not weight
• Example: A kg of matter weighs less on the moon than on Earth, but has the same mass everywhere

Length (L)

• Unit: meter (m)
• Meaning: Distance in space
• Note: Forms the basis for displacement

Time (t)

• Unit: second (s)
• Meaning: Duration of events
• Note: Used to measure rates of change

Most people are supposed to have a basic feel for these quantities; further reduction is technically impossible. Mass is perhaps the most difficult to grasp intuitively—it’s important to remember that mass refers to the amount of material, not its weight.

Weight is actually a measure of the gravitational force of attraction between a mass and a reference mass (like Earth). Thus, a kilogram of matter weighs less on the moon than on Earth, but it has the same mass.

Derived Quantities

Displacement (x)

Displacement consists of two separate (independent) pieces of information:

1. The distance between two points in space
2. The direction one must take to travel from one point to the other

Example: Saying two cities are 500 km apart provides less information than saying one city is displaced 500 km to the northeast of the other.

Quantities made up of two or more independent pieces of information are termed vectors. Displacement is a vector, whereas length alone is a scalar (a quantity that scales or adjusts size). Mass and time are also scalars.

Velocity (v)

Velocity measures the rate of change of displacement with respect to time.

Mathematical definition:

v = dx/dt

where:

• v = velocity vector
• x = displacement vector
• t = time
• dx/dt = rate of change of displacement with respect to time

Units:

• Geographic scale: kilometers per hour (km/hr)
• Voice science scale: meters per second (m/s) - the international standard

Velocity versus Speed:

• Velocity: Vector quantity including both magnitude and direction
• Speed: Scalar quantity representing only the magnitude of velocity
• Both have the same units

For air and tissue movement in voice production, velocity is typically expressed in m/s.

Momentum (p)

Momentum is derived from velocity and mass by simple multiplication or scaling.

Mathematical definition:

p = mv

where:

• p = momentum vector
• m = mass (scalar)
• v = velocity vector

Physical meaning: Momentum measures the potential force when colliding with another object.

Example: A 1,500-kg car moving at 10 m/s has momentum of 15,000 kg·m/s. This imparts the same force as a 250-kg motorcycle moving at 60 m/s because the products of mass and velocity are identical (both equal 15,000 kg·m/s).

Inertia

A body that resists a change in its momentum is said to be inertial. In other words, to change its momentum, one has to overcome its inertia, or sluggishness.

Key insight: Since every object that has mass is inertial, inertia is a universal property of materials.

Note: In the British system of units, mass is measured in “slugs,” perhaps to suggest sluggishness.

Force (f)

Force is the physical quantity imparted to an object to change its momentum—simply stated, a push or a pull.

Mathematical definition (from Newton’s second law):

f = dp/dt

where:

• f = force vector
• dp/dt = rate of change of momentum over time

Physical interpretation: Force equals the rate of change of momentum. Think of it as the ratio of a small increment in momentum over a small increment in time.

Example: The momentum of a football runner is changed by forces of collision with other players and by thrust of the runner’s feet against the ground (which the ground returns to accelerate him—Newton’s third law).

Force magnitude comparison:

• A human can bring a 6,000-kg car moving at 3 m/s to a halt in a few minutes
• A team of horses can do the same in a few seconds
• A brick wall can do it in a fraction of a second

The difference is simply the magnitude of the force that these deterrents are able to generate.

According to Newton’s third law, the force imparted to the deterrent by the car equals (but opposes in direction) the force imparted by the deterrent to the car.

Acceleration (a)

A change in momentum can be brought about by a change in mass as well as a change in velocity (or both). In typical biomechanical applications, mass tends to be conserved during motion.

In such cases, the rate of change of momentum becomes the product of a constant mass and the rate of change of velocity. Force can then be written as:

f = m(dv/dt)

Mathematical definition of acceleration:

a = dv/dt

where:

• a = acceleration vector
• dv/dt = rate of change of velocity with respect to time

Key properties:

• Acceleration is a vector quantity
• It can change in magnitude or direction or both

Examples:

• An automobile moving around a curve at constant speed is accelerated because of a change in the direction of velocity
• An automobile increasing speed along a straight line is accelerated in the magnitude of velocity

Newton’s second law (combining the above relationships):

f = ma

Force equals mass times acceleration.

Vectors Versus Scalars

All vector quantities are written with boldface symbols in technical texts to remind us of their special properties:

Vectors (boldface):

• Displacement: x
• Velocity: v
• Momentum: p
• Force: f
• Acceleration: a

Scalars (regular type):

• Mass: m
• Time: t
• Speed: |v| (magnitude of velocity)
• Length: L

Summary Table

Relevance to Voice Science

These mechanical quantities apply directly to voice production:

• Mass: Vibrating tissue mass affects frequency
• Velocity: Air and tissue velocities determine collision forces
• Force: Aerodynamic and elastic forces drive vibration
• Acceleration: Rapid tissue motion during vocal fold opening/closing
• Momentum: Tissue momentum determines impact forces

Understanding these quantities precisely enables quantitative analysis of laryngeal biomechanics.

Key Takeaways

• ✅ Three quantities are undefinable: mass (amount of material), length, and time
• ✅ Vectors contain both magnitude and direction; scalars contain only magnitude
• ✅ Velocity is rate of change of displacement; acceleration is rate of change of velocity
• ✅ Momentum equals mass times velocity, measuring potential force during collision
• ✅ Force equals mass times acceleration (Newton’s second law)
• ✅ All derived quantities ultimately trace back to mass, length, and time

Related Topics

• Subdivisions of Mechanics
• Free-Body Diagrams
• Stress and Strain

Further Reading

1. Halliday, D., & Resnick, R. (1988). Fundamentals of physics. New York: Wiley.