Physics
Rotational dynamics calculator
Moment of inertia, the parallel axis theorem, torque, Newton's second law for rotation, angular momentum and rolling. Pick the relation, leave blank the field the exercise asks for, and the formula rearranges itself.
Rotation is dynamics under other names
Every quantity of straight-line motion has a twin in rotation. Position becomes an angle, velocity an angular velocity, force a torque, mass a moment of inertia, momentum an angular momentum. The laws carry over whole: F = m·a becomes τ = I·α, and conservation of momentum becomes conservation of angular momentum. Anyone who has understood dynamics has already understood half of the rigid body.
The real difference is the moment of inertia, which unlike mass belongs not to the body alone but to the body and the axis. It depends on how much mass there is and above all on how far it is from the axis, squared. For regular shapes it is written I = k·m·r², and the factor k sums up the geometry: 1 for a ring, ½ for a disc, ⅖ for a solid sphere. If the axis does not pass through the centre of mass, add m·d² — that is the parallel axis theorem.
Rolling is the exercise where it all meets. A body rolling without slipping has to put part of the energy it gains on the way down into spinning, and the larger its k the more it spends: that is why it accelerates less than a block sliding without friction, with a = g·sin θ / (1 + k). Static friction is essential — without it the body would slide — but it does no work, because the contact point is at rest from instant to instant.
Common mistakes
- Using the moment of inertia about the centre when the axis is elsewhere: a rod turning about one end has I = mL²/3, not mL²/12.
- Working out the torque with the distance to the point of application instead of its component perpendicular to the force: you need the sine of the angle, or the true lever arm.
- Feeding revolutions per minute into formulas that want radians per second: one turn is 2π radians.
- Thinking the skater pulling in their arms conserves energy: angular momentum is conserved, and the kinetic energy goes up, paid for by the work of the muscles.
Frequently asked questions
What is the moment of inertia?
It is a body's resistance to changing its rotation, as mass is the resistance to changing velocity. It adds up every bit of mass times the square of its distance from the axis, so the same mass counts for much more when it is far out: a ring is harder to spin than a disc of the same mass and radius.
When do I use the parallel axis theorem?
When the axis of rotation does not pass through the centre of mass but is parallel to one that does. The moment of inertia grows by m·d², where d is the distance between the two axes. The classic example is the rod: mL²/12 about its centre, mL²/12 + m(L/2)² = mL²/3 about one end.
Which reaches the bottom first, a sphere or a cylinder?
The solid sphere, whatever the masses and radii. Both accelerate as g·sin θ / (1 + k), and the sphere has k = ⅖ while the solid cylinder has k = ½: the sphere puts less energy into spin and keeps more for moving down the slope. A hollow cylinder, with k = 1, comes last.
Why does a skater spin faster with the arms pulled in?
Because on the ice there is no significant external torque, so the angular momentum L = I·ω stays constant. Bringing mass closer to the axis lowers the moment of inertia, and the angular velocity rises in the same proportion. The kinetic energy goes up instead: the work is done by the arms pulling inwards.
How do I convert rpm to radians per second?
Multiply by 2π and divide by 60: one turn is 2π radians and one minute is 60 seconds. So 1 rpm is about 0.1047 rad/s and 3000 rpm about 314 rad/s. This page shows the rpm in the angular momentum summary anyway.
How this calculation works
Moment of inertia of a regular shape: I = k·m·r², with k = 1 for a ring, ½ for a disc, ⅖ for a solid sphere, ⅔ for a hollow sphere, 1/12 for a rod about its centre and ⅓ about one end (with r the length); the radius of gyration is √k·r. Parallel axis theorem: I = I_cm + m·d². Torque: τ = r·F·sin θ; solving for the angle gives the acute one, since θ and 180° − θ give the same torque. Second law for rotation: τ = I·α. Angular momentum: L = I·ω, with rotational kinetic energy ½·I·ω², revolutions per minute 60ω/2π and period 2π/ω. Conservation: I₁·ω₁ = I₂·ω₂, and since K = L²/2I the energy changes by the ratio I₁/I₂. Rolling without slipping down an incline: a = g·sin θ / (1 + k), compared with g·sin θ for a body sliding without friction; the rotational share of the energy is k/(1 + k) and the least coefficient of static friction that prevents slipping is k·tan θ/(1 + k).
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