radians per second (rad/s) to hertz (Hz) conversion

radians per second to hertz conversion table

radians per second (rad/s)hertz (Hz)
00
10.1591549430919
20.3183098861838
30.4774648292757
40.6366197723676
50.7957747154595
60.9549296585514
71.1140846016433
81.2732395447352
91.4323944878271
101.591549430919
203.1830988618379
304.7746482927569
406.3661977236758
507.9577471545948
609.5492965855137
7011.140846016433
8012.732395447352
9014.323944878271
10015.91549430919
1000159.1549430919

How to convert radians per second to hertz?

Let's explore the conversion between radians per second (rad/s) and hertz (Hz), both of which are measures of frequency.

Understanding the Conversion

Radians per second (rad/srad/s) measures angular frequency, while hertz (HzHz) measures ordinary frequency. The key difference lies in how they represent a cycle. Hertz represents the number of complete cycles per second, whereas radians per second represents the angular displacement per second. Since there are 2π2\pi radians in one complete cycle, the conversion factor is related to 2π2\pi.

Conversion Formulas

Here are the formulas to convert between radians per second and hertz:

  • Radians per Second to Hertz:

    f=ω2πf = \frac{\omega}{2\pi}

    Where:

    • ff is the frequency in Hertz (HzHz)
    • ω\omega is the angular frequency in radians per second (rad/srad/s)
  • Hertz to Radians per Second:

    ω=2πf\omega = 2\pi f

    Where:

    • ω\omega is the angular frequency in radians per second (rad/srad/s)
    • ff is the frequency in Hertz (HzHz)

Step-by-Step Conversions

Let's convert 1 rad/srad/s to hertz and 1 HzHz to radians per second.

1. Converting 1 rad/srad/s to Hertz:

Using the formula:

f=ω2πf = \frac{\omega}{2\pi}

Substitute ω=1rad/s\omega = 1 rad/s:

f=12π0.15915Hzf = \frac{1}{2\pi} \approx 0.15915 Hz

Therefore, 1 radian per second is approximately 0.15915 hertz.

2. Converting 1 Hertz to rad/srad/s:

Using the formula:

ω=2πf\omega = 2\pi f

Substitute f=1Hzf = 1 Hz:

ω=2π×16.2832rad/s\omega = 2\pi \times 1 \approx 6.2832 rad/s

Therefore, 1 hertz is approximately 6.2832 radians per second.

Interesting Facts and People

  • Heinrich Hertz (1857-1894): The unit "hertz" is named after Heinrich Hertz, a German physicist who proved the existence of electromagnetic waves. His work was crucial to the development of radio and wireless communication. You can learn more about him here.
  • The concept of radians is fundamental in physics and engineering because it simplifies many formulas, especially those involving rotational motion and oscillations.

Real-World Examples

These conversions are commonly used in various fields:

  1. Rotating Machinery: Engineers use these conversions when analyzing the speed of motors, turbines, and other rotating equipment. For example, determining the angular speed of a motor in rad/srad/s from its frequency in HzHz.
  2. Signal Processing: In signal processing, converting between frequency (HzHz) and angular frequency (rad/srad/s) is essential for designing filters and analyzing signals in the frequency domain.
  3. Acoustics: When studying sound waves, understanding the frequency content in both HzHz and rad/srad/s helps in analyzing and manipulating audio signals.
  4. Control Systems: Control engineers use these conversions to model and control systems involving oscillations or rotations, such as robotic arms or feedback loops.

See below section for step by step unit conversion with formulas and explanations. Please refer to the table below for a list of all the hertz to other unit conversions.

What is radians per second?

Radians per second (rad/s) is a unit of angular velocity or angular frequency in the International System of Units (SI). It quantifies how fast an object is rotating or revolving around an axis. Understanding radians per second involves grasping the concepts of radians, angular displacement, and their relationship to time.

Understanding Radians

A radian is a unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

  • Definition: One radian is the angle created when the length of an arc equals the radius of the circle.

  • Conversion: 2π2\pi radians is equal to 360 degrees. Therefore, 1 radian ≈ 57.3 degrees.

    1 radian=180π degrees57.31 \text{ radian} = \frac{180}{\pi} \text{ degrees} \approx 57.3^\circ

Defining Radians Per Second

Radians per second (rad/s) measures the rate of change of an angle over time. It indicates how many radians an object rotates in one second.

  • Formula: Angular velocity (ω\omega) is defined as the change in angular displacement (θ\theta) divided by the change in time (tt).

    ω=ΔθΔt\omega = \frac{\Delta\theta}{\Delta t}

    Where:

    • ω\omega is the angular velocity in rad/s.
    • Δθ\Delta\theta is the change in angular displacement in radians.
    • Δt\Delta t is the change in time in seconds.

Formation of Radians Per Second

Radians per second arises from relating circular motion to linear motion. Consider an object moving along a circular path.

  1. Angular Displacement: As the object moves, it sweeps through an angle (θ\theta) measured in radians.
  2. Time: The time it takes for the object to sweep through this angle is measured in seconds.
  3. Ratio: The ratio of the angular displacement to the time taken gives the angular velocity in radians per second.

Interesting Facts and Associations

While there isn't a specific "law" directly named after radians per second, it's a critical component in rotational dynamics, which is governed by Newton's laws of motion adapted for rotational systems.

  • Rotational Kinematics: Radians per second is analogous to meters per second in linear kinematics. Formulas involving linear velocity have rotational counterparts using angular velocity.

  • Relationship with Frequency: Angular frequency (ω\omega) is related to frequency (ff) in Hertz (cycles per second) by the formula:

    ω=2πf\omega = 2\pi f

    This shows how rad/s connects to the more commonly understood frequency.

Real-World Examples

Radians per second is used across various scientific and engineering applications to describe rotational motion:

  1. Electric Motors: The speed of an electric motor is often specified in revolutions per minute (RPM), which can be converted to radians per second. For instance, a motor spinning at 3000 RPM has an angular velocity:

    ω=3000revmin×2π rad1 rev×1 min60 s=100π rad/s314.16 rad/s\omega = 3000 \frac{\text{rev}}{\text{min}} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 100\pi \text{ rad/s} \approx 314.16 \text{ rad/s}

  2. CD/DVD Players: The rotational speed of a CD or DVD is controlled to maintain a constant linear velocity as the read head moves along the disc. This requires varying the angular velocity (in rad/s) as the read head's distance from the center changes.

  3. Turbines: The rotational speed of turbines in power plants is a crucial parameter, often measured and controlled in radians per second to optimize energy generation.

  4. Wheels: The angular speed of a wheel rotating at constant speed can be described in radians per second.

What is hertz?

Hertz (Hz) is the standard unit of frequency in the International System of Units (SI). It expresses the number of cycles of a periodic phenomenon per second. Frequency is a fundamental concept in physics and engineering, describing how often an event repeats.

Understanding Hertz

One hertz means that an event repeats once per second. A higher hertz value indicates a faster rate of repetition. This applies to various phenomena, including oscillations, waves, and vibrations.

Formation of Hertz

Hertz is a derived unit, meaning it is defined in terms of other base SI units. Specifically:

1 Hz=1 s11 \text{ Hz} = 1 \text{ s}^{-1}

This means that one hertz is equivalent to one cycle per second. The unit is named after Heinrich Rudolf Hertz, a German physicist who made significant contributions to the understanding of electromagnetic waves.

Heinrich Hertz and Electromagnetism

Heinrich Hertz (1857-1894) was the first to conclusively prove the existence of electromagnetic waves, which had been predicted by James Clerk Maxwell. He built an apparatus to produce and detect these waves, demonstrating that they travel at the speed of light and exhibit properties such as reflection and refraction. Hertz's work laid the foundation for the development of radio, television, and other wireless communication technologies. For more information about Heinrich Rudolf Hertz read his biography on Wikipedia.

Real-World Examples of Hertz

  • Alternating Current (AC): In most countries, the frequency of AC power is either 50 Hz or 60 Hz. This refers to how many times the current changes direction per second. In the United States, the standard is 60 Hz.

  • CPU Clock Speed: The clock speed of a computer's central processing unit (CPU) is measured in gigahertz (GHz). For example, a 3 GHz processor completes 3 billion cycles per second. This clock speed governs how quickly the CPU can execute instructions.

  • Radio Frequencies: Radio waves are electromagnetic waves used for communication. Their frequencies are measured in hertz (Hz), kilohertz (kHz), megahertz (MHz), and gigahertz (GHz). For example, FM radio stations broadcast in the MHz range, while mobile phones use GHz frequencies.

  • Audio Frequencies: The range of human hearing is typically between 20 Hz and 20,000 Hz (20 kHz). Lower frequencies correspond to bass sounds, while higher frequencies correspond to treble sounds. Musical instruments produce a range of frequencies within this spectrum.

  • Oscillators: Oscillators are electronic circuits that produce periodic signals. Their frequencies are measured in hertz and are used in various applications, such as clocks, timers, and signal generators. The frequency of an oscillator determines the rate at which it produces these signals.

Interesting Facts

  • Prefixes are commonly used with hertz to denote larger frequencies:

    • 1 kHz (kilohertz) = 1,000 Hz
    • 1 MHz (megahertz) = 1,000,000 Hz
    • 1 GHz (gigahertz) = 1,000,000,000 Hz
  • The inverse of frequency (1/f) is the period (T), which is the time it takes for one complete cycle to occur. The period is measured in seconds.

T=1fT = \frac{1}{f}

Complete radians per second conversion table

Enter # of radians per second
Convert 1 rad/s to other unitsResult
radians per second to millihertz (rad/s to mHz)159.1549430919
radians per second to hertz (rad/s to Hz)0.1591549430919
radians per second to kilohertz (rad/s to kHz)0.0001591549430919
radians per second to megahertz (rad/s to MHz)1.591549430919e-7
radians per second to gigahertz (rad/s to GHz)1.591549430919e-10
radians per second to terahertz (rad/s to THz)1.591549430919e-13
radians per second to rotations per minute (rad/s to rpm)9.5492965855137
radians per second to degrees per second (rad/s to deg/s)57.295779513082