A uniform solid sphere and a thin hoop with the same mass and radius are released simultaneously from rest on slopes with the same angle of inclination.
Adjust conditions:
The sliders adjust the values
球が坂に沿って1 m進んだ瞬間、輪は何m進んでいるか？
摩擦のない坂で同時に放すと、同じ距離に達するのはどちらが先か？
The equations explain the phenomenon
1The rotational equation follows from the equations of motion of the particles.dLdt=N……
2The component of a rigid body's angular momentum along its rotation axis is calculated.Lz=I ω，I=Σ mi ρi²……
3The rotational equation is compared with the equation of motion.I dωdt=Nz……
4Rolling without slipping is described by 2 equations of motion.M a=M g sinθ− f，I α=f R……
The condition for rolling without slipping is that the contact-point velocity is 0 → the center-of-mass speed satisfies v = Rω → differentiation with respect to time gives a = Rα.
5The condition for rolling without slipping eliminates the frictional force from the equations.a=g sinθ1 + I/MR²……
6The condition for rolling without slipping is checked.μs≧k tanθ1 + k……
7The distances traveled by the sphere and hoop are compared at the same time.xhoopxsphere=ahoopasphere=(1 + 25)/(1 + 1)=0.70……
Angular momentum now explains other phenomena.
8Energy is used to compare the speeds after descending through the same height.M g h=½ M v²+½ I ω²=½ M v² (1 + IMR²)……
9Conservation of angular momentum determines how fast a skater spins after pulling their arms inward.I₁ ω₁=I₂ ω₂……
10Why a tilted top's axis rotatesΩ≒M g d/(I ω)……
11The result is reconsidered as a conservation law.N=0⇒L=constant……
What this chapter has shown
During rolling without slipping, the hoop's mass is distributed farther from the axis than the sphere's, giving it a larger I/MR². Its center-of-mass acceleration g sinθ/(1+I/MR²) is therefore smaller. After descending through the same height, the hoop also has a larger fraction of its kinetic energy in rotation.
0.70 timesDistance traveled by the hoop ÷ distance traveled by the sphere
However, when released from rest on a frictionless slope, both the sphere and the hoop slide without rotating and arrive simultaneously.
In everyday lifeReducing the mass near the rim of a bicycle wheel makes it easier to accelerate from rest.
Detailed summary
Rotational motion can be analyzed using equations of the same form as the equation of motion, with moment of inertia replacing mass and torque replacing force. Rolling speed depends only on shape, and angular momentum is conserved when no external torque acts. A skater's spin, the precession of a spinning top, and a planet's areal velocity can all be derived from 1 common equation.