Analyze circular and rotational dynamics. Calculate torque, angular velocity, and moment of inertia.
🔥 Popular Calculations
Calculate the rate of change of angular displacement over time.
Open CalculatorCalculate the turning effect or rotational force acting on an object.
Open CalculatorCalculate the moment of inertia for various shapes and mass distributions.
Open CalculatorCalculate the force that keeps an object moving in a circular path.
Open CalculatorCalculate the angle through which an object or point has gathered rotation.
Open CalculatorAnalyze parameters of uniform circular motion including velocity and acceleration.
Open CalculatorCalculate the period of oscillation for a simple pendulum.
Open CalculatorCalculate angular frequency in radians per second.
Open CalculatorCalculate the rotational momentum of a rotating object.
Open CalculatorCalculate the G-force and relative centrifugal force in a centrifuge.
Open CalculatorCalculate the amount of energy stored in a rotating flywheel.
Open CalculatorCalculate the polar moment of inertia (torsional constant) for shafts.
Open CalculatorCalculate the energy possessed by an object due to its rotation.
Open CalculatorCalculate the rotational stiffness or torsion constant of a system.
Open CalculatorRotational motion is the study of objects spinning on an axis. In introductory physics, we often start with linear motion (sliding a block down a ramp), but the real world is full of rotation. Wheels, planets, gears, and electrons all spin. Rotational dynamics uses the same fundamental laws as linear dynamics, but with a twist—literally.
Every linear quantity has a rotational analog. Position becomes angle (θ). Velocity becomes angular velocity (ω). Acceleration becomes angular acceleration (α). Force becomes torque (τ), and mass becomes moment of inertia (I).
Understanding rotational motion allows engineers to balance tires on a car, design stable satellites using gyroscopes, and optimize the efficiency of wind turbines. The calculators below help you solve for these specific variables, but seeing the bigger picture of how they relate is key to mastering mechanics.
Torque (τ) is the measure of how much a force acting on an object causes that object to
rotate. It depends on three things:
1. The magnitude of the force applied.
2. The distance from the pivot point (radius or lever arm).
3. The angle at which the force is applied (perpendicular is best).
"Give me a lever long enough and a fulcrum on which to place it, and I shall move the
world." — Archimedes
This is the rotational equivalent of mass. It measures how difficult it is to change an object's rotation. Unlike mass, it depends on shape! A hollow ring has a higher moment of inertia than a solid disk of the same mass because the mass is distributed further from the center.
Just as moving objects have linear momentum (p=mv), spinning objects have angular momentum (L=Iω). It is a conserved quantity. This is why a figure skater spins faster when she pulls her arms in—her moment of inertia (I) decreases, so her angular velocity (ω) must increase to keep L constant.
Centripetal Force: This is the "real" force pulling an object toward the
center of a circle (e.g., gravity holding the moon).
Centrifugal Force: This is a "fictitious" force experienced by the observer
inside the rotating system. It feels like you are being pushed outward, but it's
really just your own inertia trying to travel in a straight line while the car turns.
The beauty of physics is in its symmetry. If you know Newton's Second Law for linear motion (F=ma), you automatically know it for rotation (τ=Iα).
Flywheels stores energy rotationally. They are used in engines to smooth out power pulses and in hybrid buses to capture braking energy (Kinetic Energy Recovery Systems). They rely on high moments of inertia and high speeds.
Helicopters need a tail rotor because of conservation of angular momentum. As the main blades spin one way, the body of the helicopter wants to spin the other way (Newton's 3rd Law/Torque). The tail rotor provides a counter-torque to keep it steady.
Your phone knows when you rotate it from portrait to landscape thanks to micro-electromechanical (MEMS) gyroscopes. These sensors detect angular velocity and allow for screen rotation and motioncontrolled gaming.
Yes! Just as Work = Force × Distance for linear motion, Work = Torque × Angle (in radians) for rotational motion. Power is Torque × Angular Velocity. This is why car engines are rated in both Torque (force of twist) and Horsepower (rate of work).
Radians are "natural" units. One radian is the angle where the arc length equals the radius. Many physics formulas (like v = rω or a = rα) ONLY work if you use radians. If you use degrees, you have to add messy conversion factors (π/180) everywhere.
Absolutely. If you push a block right through its center of mass, it accelerates linearly but doesn't rotate (Zero Torque, Net Force). Conversely, if you spin a wheel by pushing up on one side and down on the other with equal force, it spins but stays in place (Net Torque, Zero Net Force).
Conservation of angular momentum makes the spinning wheels resistant to tilting (changing the direction of their angular momentum vector). This stabilization effect helps, though steering geometry (trail) is arguably even more important for bike stability.
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