Chapter 6 Lectures: Applying Newton's Laws: Period and Frequency

Period in Rotational and Periodic Motion

• Symbolic Notation and Physics Conventions:

  • The period of rotation is represented by the uppercase symbol $T$.

  • Physics frequently reuses identical symbols across different subfields, requiring attention to context:

    • The symbol $T$ denotes period in rotational and oscillatory motion, tension force in mechanics, and temperature in thermodynamics.

    • The symbol $N$ denotes the normal force in free-body diagrams as well as Newtons ($\text{N}$), the SI unit of force.

    • Lowercase $t$ universally represents arbitrary or elapsed time, whereas capital $T$ strictly designates the period of a repeating cycle.

    • To accommodate the vast quantity of distinct physical variables, physics incorporates Greek letters, superscripts, and subscripts.

• Formal Definition of Period:

  • The period of rotation ($T$) is defined as the time it takes for a rotating object to complete one full revolution or one full cycle.

  • In uniform circular motion, an object travels along a fixed circular path at a constant speed; the period represents the time required to complete the trajectory and return to the exact initial position.

• Generalization to Oscillating and Vibrating Systems:

  • A cycle represents any closed, repetitive pathway.

  • Periodic motion is defined as motion that repeats itself cyclically over regular, identical time intervals.

  • In back-and-forth vibrational or oscillatory motion, the period is the time taken for the system to displace from an initial starting position, travel through its motion, and return back to that identical starting position.

Circular Motion and the Circumference-Speed Relationship

• Kinematic Speed in Uniform Circular Motion:

  • The term uniform indicates that the magnitude of velocity (speed) remains constant throughout the motion, even though the direction of velocity changes continuously.

  • Standard average speed is defined as the scalar ratio of distance covered to elapsed time:         $\text{speed} = \frac{\text{distance}}{\text{time}}$

• Derivation of Orbital Speed:

  • For one full cycle around a circular trajectory, the distance traversed is equal to the circumference ($C$) of the circle.

  • The circumference of a circle of radius $r$ is defined geometrically as:         $C = 2\pi r$

  • The time required to traverse this circumference is the period ($T$).

  • Speed ($v$), represented as a scalar quantity without a vector arrow, is expressed as:         $v = \frac{2\pi r}{T}$

• Geometric Definition and Significance of Pi ($\pi$):

  • Historically, constructing a physical circle required cutting straight segments into progressively smaller, finite pieces and smoothing them out.

  • A circle can geometrically be modeled as a regular polygon with an infinite number of infinitesimal sides.

  • For any circle, the diameter ($d$) is the line segment passing through the center connecting two opposing points, equal to two radii ($d = 2r$).

  • The irrational constant $\pi$ is defined as the exact ratio of a circle's circumference to its diameter:         $\pi = \frac{C}{d}$

  • Historically, calculating the ratio of measured circumference to measured diameter served as a quantitative metric to verify the geometric precision and perfection of circular manufacturing.

  • Rearranging this fundamental definition yields the standard formulation for circumference:         $C = \pi d = 2\pi r$

The Nature and Definition of Frequency

• Conceptual Overview of Frequency:

  • Frequency is fundamentally a counting measurement that quantifies how often a particular repeating event occurs within a specified duration of time.

  • Colloquial analogy: Visiting a restroom $7$ times throughout an entire day represents a frequency of $7\,\text{times per day}$, where the day serves as the unit of time.

• Mathematical and Physical Definition:

  • Frequency ($f$) is defined as the number of full revolutions or cycles completed by an object per unit time.

  • Symbolic distinction: Lowercase $f$ denotes frequency; unlike the frictional force vector ($\vec{f}$ or $f$), frequency is a scalar value and does not feature a vector arrow.

  • Frequency is the exact inverse (reciprocal) of the period ($T$):         $f = \frac{1}{T}$

  • Dimensionally, the numerator represents one completed cycle ($1\,\text{cycle}$) and the denominator represents the time needed per cycle ($T$):         $\text{Dimensions of } f = \frac{1}{\text{time}}$

Units of Frequency and Physical Applications

• The Hertz ($\text{Hz}$):

  • The SI unit of frequency is the Hertz ($\text{Hz}$).

  • One Hertz is defined as the completion of exactly one full cycle or revolution in one second:         $1\,\text{Hz} = 1\,\text{cycle per second} = 1\,\text{s}^{-1}$

  • Because cycle counts are dimensionless numbers, the base SI unit dimension is inverse seconds ($\text{s}^{-1}$).

• Electromagnetic Oscillations and Radio Transmission:

  • Radio signals consist of electromagnetic waves that oscillate continuously up and down transverse to their direction of propagation.

  • One complete up-and-down oscillation represents one cycle.

  • A frequency of $1\,\text{Hz}$ corresponds to a wave oscillating through one full cycle per second, which represents a low frequency.

  • Commercial broadcast frequencies operate at vastly higher scales, often utilizing megahertz ($\text{MHz}$), where the metric prefix mega denotes one million ($10^6$):

    • Example: A radio station broadcasting at $96.7\,\text{MHz}$ (such as KCAL) has a frequency of:             $96.7 \times 10^6\,\text{Hz} = 96{,}700{,}000\,\text{Hz}$

    • This value means the transmitted electromagnetic wave undergoes $96{,}700{,}000$ complete oscillations every second.

• Rotational Mechanical Frequency (RPM):

  • Mechanical engineering commonly expresses rotational frequency in revolutions per minute ($\text{RPM}$).

  • In internal combustion engines, the motor rotates rapidly; the combustion strokes are timed precisely to impart successive rotational boosts to keep the motor spinning.

  • This rotational motion is mechanically transmitted through the axles to turn the vehicle's wheels.

  • While revolutions per second ($\text{rev/s}$) equals Hertz ($\text{Hz}$), $\text{RPM}$ quantifies the number of full rotations completed in a $60\,\text{second}$ interval.

Earth's Orbital Frequency and Fractional Rates

• Orbital Period Parameters:

  • The Earth's annual revolution around the Sun constitutes a uniform periodic orbital cycle.

  • The time required for the Earth to complete one full revolution is one year.

  • Expressed to three significant figures, the period of the Earth's orbit is:         $T = 365\,\text{days}$

• Astronomical Observations of Orbit:

  • During Earth's orbit, the hemisphere facing the Sun experiences daylight, whereas the hemisphere facing away into space experiences nighttime.

  • As Earth progresses along its circular trajectory, the nocturnal field of view points toward different regions of space, altering the constellations visible each month.

  • These monthly variations in visible constellations form the historical basis for zodiac constellations and symbols.

• Calculation of Orbital Frequency:

  • Using the reciprocal relationship between period and frequency:         $f = \frac{1}{T}$

  • Substituting the orbital period of $365\,\text{days}$ yields:         $f = \frac{1}{365}\,\text{day}^{-1}$

  • Written in terms of inverse years, the frequency is simply:         $f = 1\,\text{year}^{-1}$

• Concept of Fractional Frequencies:

  • Frequency measurements are not limited to integer counts; they can be fractional.

  • Over a duration of one single day (the time needed for Earth to rotate once on its internal axis), the Earth does not complete an entire orbit around the Sun.

  • Instead, the Earth traverses a fractional segment of its total orbital circumference equal to:         $\frac{1}{365}\text{ of the full circumference}$

  • Evaluating orbital frequency in units of seconds would yield an even smaller fractional count per unit time, demonstrating that frequency quantifies completed or partial cycle progression relative to any selected time base.