Rolling kinetic energy is a sum of checked rotational and translational parts. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Matched translation and rotation give eighteen joules

Rotational energy 9 J plus translational energy 9 J gives total kinetic energy 18 J.

9+9=189+9=18
Rolling energy partition rowTranslation and rotation add into the total kinetic energy.omega=3 1/sKrot=9 JKtrans=9 JKtotal=18 J

Faster rolling squares both energy ledgers

Rotational energy 16 J plus translational energy 16 J gives total kinetic energy 32 J.

16+16=3216+16=32
Rolling energy partition rowTranslation and rotation add into the total kinetic energy.omega=4 1/sKrot=16 JKtrans=16 JKtotal=32 J

Heavier matched rolling doubles the base total

Rotational energy 18 J plus translational energy 18 J gives total kinetic energy 36 J.

18+18=3618+18=36
Rolling energy partition rowTranslation and rotation add into the total kinetic energy.omega=3 1/sKrot=18 JKtrans=18 JKtotal=36 J