What is rotational motion?
Rotation is the kinematics and dynamics of things that spin — wheels, gears, planets, molecules, hurricanes. The bookkeeping parallels the linear version almost line-for-line: linear displacement becomes angular displacement θ (radians), linear velocity becomes angular velocity ω (rad/s), linear acceleration becomes angular acceleration α (rad/s²), mass becomes moment of inertia I (kg·m²), and force becomes torque τ (N·m). Newton's second law takes its rotational shape as τ = I·α, and angular momentum L = I·ω is conserved in the absence of external torque, exactly as linear momentum is conserved in the absence of external force. The six calculators on this panel cover the working formulas: centripetal force for circular motion, angular-velocity unit conversion, torque, moment of inertia for common shapes, angular momentum with rotational kinetic energy, and the centrifugal pseudo-force in rotating frames.
Two parallelisms to remember: linear momentum p = mv and angular momentum L = Iω; force F = ma and torque τ = Iα. The moment of inertia I plays the same role as mass m, but it depends on shape and how the mass is distributed relative to the rotation axis.
Centripetal force — the inward pull that bends the path
Any object moving in a circle is changing direction, which is acceleration, which requires force — directed toward the center. The magnitude is F = mv²/r = mω²r. The default sample (m = 1 kg, v = 10 m/s, r = 2 m) gives F = 50 N and centripetal acceleration 50 m/s² (≈ 5 g), with angular velocity 5 rad/s (≈ 47.75 RPM). The force is real — for a ball on a string, it is the string tension; for a car in a turn, it is the friction between tires and road; for an orbiting satellite, it is gravity. Take the force away (string snaps, tires lose grip) and the object flies off tangentially, not radially outward: that outward push you feel in a turning car is the centrifugal pseudo-force in the rotating frame, not the centripetal force in the inertial frame.
Angular velocity — the four units that meet here
The same spinning object can be described in revolutions per minute (RPM), in radians per second (rad/s, the SI unit), in revolutions per second (Hz), or in degrees per second. The default sample (1200 RPM) becomes ω ≈ 125.66 rad/s, 7,200 deg/s, or 20 Hz, with a period of 0.05 s per revolution. The relations: ω = 2πf = 2π n/60 where n is RPM; the factor 2π converts revolutions into radians, and 60 converts minutes into seconds. These unit conversions matter less for textbook work than for engines and motors, where RPM is the dial reading and torque (in N·m) is the working output.
Torque — the lever, weighted by the angle
Torque is the rotational analogue of force: τ = r·F·sinθ, where r is the lever arm length, F is the applied force, and θ is the angle between them. The default sample (F = 50 N, r = 0.3 m, θ = 90°) gives τ = 15 N·m. The sinθ factor means the force is most effective at 90° (perpendicular to the lever); pushing straight along the lever (θ = 0 or 180°) produces zero torque, no matter how hard you push. This is why a wrench turns a bolt most easily when you push perpendicular to the handle, and why a longer handle gives more torque at the same applied force — which is also why wrenches, doorknobs, and steering wheels are larger than the axles they turn.
Moment of inertia — the rotational mass
Moment of inertia is the rotational analog of mass, and it depends on how the mass is distributed about the axis. For a solid sphere of mass m and radius R, I = (2/5)mR²; for a solid disk or cylinder about its central axis, I = (1/2)mR²; for a thin ring or hoop, I = mR²; for a uniform rod about its center, I = (1/12)mL². The default sample (solid sphere, m = 2 kg, R = 1 m) gives I = 0.8 kg·m². The practical consequence is that a hoop is harder to spin than a disk of the same mass and radius (all the mass sits at the rim), and a hollow sphere is harder to spin than a solid one. The further the mass is from the axis, the larger I — which is why a figure skater pulling in their arms speeds up: I decreases, ω increases, and L = Iω stays constant.
Angular momentum — the conserved quantity
Angular momentum is L = Iω, in kg·m²/s; the rotational kinetic energy is KE = ½Iω². The default sample (I = 5 kg·m², ω = 10 rad/s) gives L = 50 kg·m²/s and KE = 250 J. The conservation law is the same as for linear momentum: in the absence of external torque, L stays constant. A spinning top stays spinning for as long as friction (an external torque) stays small; a freely orbiting planet conserves angular momentum by sweeping equal areas in equal times (Kepler's second law). The combined statement is why gyroscopes resist tilting: changing the direction of L requires a torque, and a fast-spinning gyro has so much L that small torques barely deflect it.
Centrifugal force — the rotating-frame pseudo-force
Inside a rotating frame, an object at rest feels a fictitious outward force equal to the centripetal force it would need to keep it on the circle. The magnitude is F = mω²r, identical to the centripetal formula. The default sample (m = 0.1 kg, ω = 100 rad/s, r = 0.15 m) gives F = 150 N outward, a tangential speed of 15 m/s, and a g-force of ~152.91 g. The pseudo-force does not exist in an inertial (non-rotating) frame, where the inward force from the rope or wall is the centripetal force and there is no outward push at all. Centrifugal is what makes water stay in the bucket during a vertical swing, what spins clothes dry in a washing machine, what separates blood components in a centrifuge, and what holds passengers against the outer wall of a rotating space-station design. The two forces are equal in magnitude and opposite in direction by definition.
Common misconceptions
- Centrifugal force is a real force pushing you outward. It is a fictitious force that appears only in a rotating (non-inertial) frame. In an inertial frame, the only force is the inward centripetal one — what you feel as "outward" is your inertia resisting the inward pull.
- Angular momentum keeps a top spinning forever. Angular momentum is conserved only in the absence of external torque. A top slows because friction at the tip is an external torque; a frictionless bearing (a hard vacuum, a superconductor) can keep a wheel spinning for days, but never forever.
- Heavier objects have larger moments of inertia. Heavier objects of the same shape do; but a hollow hoop and a solid disk of the same mass can have very different moments, because it is the distribution that matters, not just the total mass.
- An object in circular motion has no net force. It has zero net force if moving at constant speed in a straight line. A circular path is constant acceleration toward the center, requiring a constant inward force. Take the force away and the object leaves the circle.
Related tools: Mechanics for the linear dynamics underneath, Fluids for vortices and rotating flow, and Modern Physics for the quantum angular momentum that shows up in atoms.
