Why a Figure Skater Spins Faster With Her Arms Pulled In

Every real object has finite size, yet a great deal of physics gets away with treating objects as single points -- a shortcut that stops working the moment rotation matters. A rigid body, an idealised object whose particles never change their distances from one another, turns out to have only a small number of genuinely possible motions. Sliding a block down a ramp without any sideways motion is pure translation, every single particle of the block sharing exactly the same velocity at every instant. Rolling a cylinder down that same ramp looks similar at first glance, but is fundamentally different: different points on the cylinder move at different speeds at any given instant (the point touching the ramp is briefly at rest, if the cylinder rolls without slipping), so rolling is NOT pure translation -- it is translation combined with something else. That 'something else' is revealed by constraining a rigid body so it cannot translate at all, fixing it along a line: the only motion left is rotation about that fixed axis, and every single particle of the body then moves in its own circle, lying in a plane perpendicular to the axis, centred on the axis -- a ceiling fan, a potter's wheel, a spinning door are everyday examples. Rolling motion, then, is precisely this rotation about a (moving) axis, combined with the translation of that axis itself -- exactly the insight that lets a cylinder's complicated-looking motion be split into two much simpler pieces.

For a system of particles, the centre of mass is the mass-weighted average position, R = (Σ mi ri)/M, generalising directly to a continuous body by turning the sum into an integral. For any object with a natural symmetry, the centre of mass is easy to find without any calculation: a uniform rod, ring, disc, or sphere all have their centre of mass exactly at their own geometric centre, since for every small mass element on one side there is an identical element positioned symmetrically opposite. The genuinely powerful result appears once this point's own motion is examined: differentiating R twice with respect to time and applying Newton's second law to every particle individually shows that MA = Fext, where A is the acceleration of the centre of mass and Fext is the vector sum of ONLY the external forces (every internal force between particles cancels out in pairs, by Newton's third law). This means the centre of mass of any system, however complicated its internal motion, moves EXACTLY as if the system's entire mass were concentrated at that one point and every external force acted there directly -- a projectile that explodes mid-flight has its fragments scatter in complicated directions, yet their combined centre of mass continues along the exact same parabolic path the intact projectile would have followed, since the explosion's forces are entirely internal.

Beyond the scalar product already used for work, there is a genuinely different way to multiply two vectors, one that produces another VECTOR rather than a plain number: the vector (or cross) product, written a x b, with magnitude ab sin θ (θ being the angle between a and b) and a direction perpendicular to BOTH a and b, determined by the right-hand rule (curl the fingers of the right hand from a toward b; the thumb points along a x b). This product behaves very differently from ordinary multiplication: it is NOT commutative, since reversing the order reverses the direction, a x b = -(b x a), though the magnitude stays the same either way; it IS distributive over addition, a x (b+c) = a x b + a x c; and a vector crossed with itself is always zero, since sin 0° = 0. Written in components using the unit vectors î, ĵ, k̂ (which satisfy î x ĵ = k̂, ĵ x k̂ = î, k̂ x î = ĵ, cycling in order, and the reverse cyclic products carry a negative sign), the vector product expands into a determinant-style formula that is straightforward to compute for any two vectors given in component form. Just as the scalar product was the perfect tool for work (a quantity with no natural direction), the vector product turns out to be exactly the right tool for two of this chapter's central quantities, torque and angular momentum, both of which genuinely need a direction (along the rotation axis) that a plain number could never capture.

Every particle of a rotating rigid body sweeps out the same angle in the same time, so angular velocity, ω = dθ/dt, describes the WHOLE body at once, not just one particle -- and it turns out to genuinely be a vector, directed along the rotation axis by the right-hand rule, connected to any one particle's ordinary linear velocity by v = ω x r, a direct application of the vector product. Force has a rotational analogue too: applying a given force to a door works far better far from the hinge than close to it, and pushing perpendicular to the door works better than pushing along it -- this combined dependence on both distance and direction is captured exactly by torque (or moment of force), τ = r x F, another vector product, vanishing completely if the force passes through the axis or acts parallel to r. Momentum's own rotational analogue is angular momentum, l = r x p for a single particle, extended to a whole system by vector-summing every particle's contribution; differentiating l with respect to time, using the product rule and Newton's second law, proves dl/dt = τ, the exact rotational mirror of F = dp/dt. Extending this to a full system of particles, and using Newton's third law to show that every INTERNAL torque cancels out in equal-and-opposite pairs, leaves only the external torques: dL/dt = τext -- and immediately, when the total external torque is zero, L stays exactly constant, the conservation of angular momentum, the rotational twin of the conservation of linear momentum already familiar from a system with zero net external force.

A single particle is in equilibrium whenever the net force on it is zero -- but an extended rigid body needs a genuinely second, independent condition: zero net force (translational equilibrium) AND zero net torque (rotational equilibrium), since a body can satisfy either one without the other. A rod pushed by two equal, opposite, parallel forces at its two ends has zero net force (it will not accelerate), yet a nonzero net torque, since both forces try to rotate it the same way -- this is a couple, and a couple can spin an unfixed, unsupported object with absolutely no net force acting on it at all, exactly what happens turning a bottle cap or a compass needle aligning with the Earth's magnetic field. An ideal lever, a light rod pivoted at a fulcrum, balances exactly when the moments of the two forces about the fulcrum are equal and opposite: d1F1 = d2F2, the principle of moments, giving a mechanical advantage F1/F2 = d2/d1 identical in spirit to the simple-machines result from the Work, Energy and Power chapter, but now derived directly from the rotational-equilibrium condition rather than from energy conservation. The centre of gravity, the single point where an extended body's ENTIRE weight can be treated as acting, is defined as the point where the total gravitational torque on the body is exactly zero -- and coincides exactly with the centre of mass whenever gravity does not vary meaningfully across the body's own size, which is true for essentially every everyday object.

Mass measures how strongly an object resists a change to its linear motion -- rotation has its own exact analogue. Adding up the kinetic energy of every individual particle of a body rotating about a fixed axis at angular speed ω (each particle at distance ri moving with speed ωri) gives a total K = (1/2)ω²(Σ mi ri²); defining the moment of inertia as I = Σ mi ri², this collapses to the strikingly clean K = (1/2)Iω², the exact rotational mirror of the familiar K = (1/2)mv². Unlike ordinary mass, moment of inertia is NOT a fixed property of an object alone -- it depends entirely on how that mass is distributed relative to the specific axis chosen: a thin ring of mass M and radius R spinning about its own centre has I = MR² (every particle sits at the same distance R), while the same mass arranged as a rod of length l with two point masses at its ends, rotating about its centre, gives I = Ml²/4 instead -- a completely different value for the identical total mass, purely because the mass is distributed differently relative to the axis. Standard shapes (rings, discs, cylinders, spheres, rods) about their natural symmetry axes have moments of inertia that follow clean, well-known formulas (a solid sphere about its diameter, for instance, is (2/5)MR²), letting the radius of gyration, k (defined by I = Mk², the distance at which the body's whole mass COULD be concentrated to give the same I) summarise a shape's rotational resistance in a single number. This is not merely academic: engines deliberately fit a heavy flywheel, a disc with a large moment of inertia, specifically because its rotational inertia resists sudden speed changes, smoothing out an engine's inherently jerky power delivery into a comfortable, steady ride.

Angular displacement (θ), angular velocity (ω = dθ/dt), and angular acceleration (α = dω/dt) map so exactly onto ordinary linear displacement, velocity, and acceleration that the entire set of familiar kinematic equations for constant acceleration carries straight across, symbol for symbol: ω = ω0 + αt, θ = θ0 + ω0t + (1/2)αt², and ω² = ω0² + 2α(θ-θ0), each one identical in form to its linear counterpart, simply written with the rotational letters in place of the linear ones. This is not a coincidence dressed up to look clever -- the underlying mathematics (a quantity, its rate of change, and the rate of change of THAT rate) is genuinely the same problem, applied to a different physical variable, and the exact same calculus produces the exact same equations. A motor wheel spinning up from a known initial angular speed to a known final angular speed over a measured time interval can therefore be analysed using precisely these three equations, finding its angular acceleration and the total angle (hence number of full revolutions) it turns through, exactly as a car's linear speed, acceleration and distance travelled would be found using the ordinary versions of these same three equations.

The rotational dynamics of a fixed-axis system follow directly from summing every particle's own contribution to torque and kinetic energy: the work done by a torque τ turning a body through a small angle dθ is dW = τ dθ, the exact rotational mirror of dW = F ds; dividing by dt gives instantaneous power P = τω, mirroring P = Fv exactly. Setting the rate of work done equal to the rate of increase of rotational kinetic energy, d[(1/2)Iω²]/dt = Iωα, and equating this to τω, gives the single most important result in the whole chapter: τ = Iα, the direct rotational twin of F = ma, showing that torque produces angular acceleration in exactly the same proportional way that force produces ordinary acceleration, with moment of inertia playing the role mass always played. Every single quantity from linear mechanics now has a confirmed rotational partner: displacement becomes angle, velocity becomes angular velocity, mass becomes moment of inertia, force becomes torque, momentum becomes angular momentum, and Newton's second law itself, F = ma, becomes τ = Iα -- a complete, working, parallel universe of rotational mechanics, built entirely from the vector product and careful bookkeeping across every particle of a rigid body.

For a body rotating about a fixed axis with a natural symmetry (a wheel, a disc, a sphere spinning about its own axis of symmetry), the whole system's angular momentum reduces to the strikingly simple L = Iω -- so if the total external torque on such a body is zero, Iω stays exactly, permanently constant. Since I depends only on how mass is distributed relative to the axis, and this distribution can genuinely be changed WITHOUT applying any external torque, a spinning object can therefore change its own angular speed simply by changing its own shape. A figure skater or a spinning acrobat begins a spin with arms outstretched, a comparatively large moment of inertia, then pulls the arms in close to the body, shrinking I dramatically -- and since Iω must stay exactly constant with no external torque acting (ignoring the small friction at the point of contact), ω must increase to compensate, exactly why pulling the arms in makes the spin visibly, dramatically faster. The same principle, demonstrated on an ordinary swivel chair with a person holding weights in outstretched arms, or performed by divers tucking into a tight ball mid-air to spin faster before straightening out again to slow down just before entering the water, is genuinely the SAME physics as a distant binary star system or a planet in an elliptical orbit speeding up as it swings closer to its companion -- conservation of angular momentum, quietly running underneath an enormous range of spinning, twirling, and orbiting motion, from a playground to the outer solar system.