Angular Velocity

Online calculator and formulas for calculating angular velocity

Angular Velocity Calculator

Rotational motion

Calculates the relationship between angular velocity (ω), angle (φ) and time (t) in rotational motions.

Result
Angular velocity:
Angle:
Time:

Example Calculation

Example: Machine shaft
Problem:

A machine shaft rotates through 540° in 3 seconds. What is the angular velocity?

Given:
  • Angle φ = 540° = 3π rad
  • Time t = 3 s
  • Find: Angular velocity ω
Solution:

1. Convert degrees → radians:

\[\phi = 540° \times \frac{\pi}{180°} = 3\pi \text{ rad}\]

2. Calculate angular velocity:

\[\omega = \frac{\phi}{t}\]
\[\omega = \frac{3\pi \text{ rad}}{3 \text{ s}} = \pi \text{ rad/s}\]
\[\omega = 3.14 \text{ rad/s}\]
Practical applications
Mechanical engineering: Motor control, gears, torque
Robotics: Joint control, motion planning
Physics: Gyrocompass, satellite navigation
Understanding angular velocity

Angular velocity: Describes how fast the angle changes. It is the rotational equivalent to linear velocity and is measured in rad/s.

Formulas for angular velocity

Angular velocity is the rotational equivalent to linear velocity. It describes the change of angle per unit of time in rotational motions.

Calculate angular velocity

Basic formula of rotational mechanics: angle change per time.

\[\omega = \frac{\phi}{t}\]
ω = Angular velocity [rad/s]
φ = Angle [rad]
t = Time [s]
Calculate angle

Angle from angular velocity and time.

\[\phi = \omega \times t\]
For constant angular velocity.
Calculate time

Required time for a certain angular rotation.

\[t = \frac{\phi}{\omega}\]
Important for timing calculations in automation.
Relationship to linear velocity

Connection between linear and angular velocity.

\[v = \omega \times r\]
v = Linear velocity [m/s]
r = Radius [m]
Units and conversions
Angle:
1 full circle = 360° = 2π rad
1 rad ≈ 57.3°
Frequency:
ω = 2πf
f = Rotational speed [Hz]
RPM:
n [RPM] = 30ω/π
ω = πn/30

Detailed description of angular velocity

Physical Fundamentals

Angular velocity is a fundamental quantity in rotational mechanics. It describes how fast a body rotates around an axis and is defined as the change of angle per unit of time.

Unlike linear velocity, which describes a direction in space, angular velocity is a vector quantity that points along the rotation axis.

Usage Instructions

Select with the radio buttons which quantity should be calculated. Pay special attention to the angle units (degrees vs. radians).

Application Areas

Mechanical Engineering

Motor control, gears, torque calculation. Basis for drive technology and control engineering.

Robotics and Automation

Joint control, motion planning, servo drives. Precise positioning and speed control.

Aerospace

Gyrocompass, gyroscopes, satellite orientation. Navigation and stabilization of aircraft.

Angular velocity in practice

Angular velocities can be found everywhere in technology. Here are typical values from different applications:

Everyday devices

Clock hand: ~0.1 rad/s
Fan: ~100 rad/s
Hard drive: ~500 rad/s

Vehicles

Car wheel: ~80 rad/s
Motorcycle: ~600 rad/s
Turbine: ~3000 rad/s

Precision technology

Gyroscope: ~1000 rad/s
Centrifuge: ~10000 rad/s
Atomic nucleus: ~10²³ rad/s

Memory aid: One radian corresponds to about 57.3°. For a full circle rotation (360°), 2π ≈ 6.28 radians are needed.


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