Rolling
Đã giới thiệuA race down a ramp: which wheel wins?
- Roll a solid disk and a hollow ring down the same ramp. They start together — but the disk wins every time.
- A rolling object is doing two things at once: moving forward and spinning.
- Its energy is shared between the two, and the split depends on the shape.
- Understanding that split is the key to rolling motion.
Two kinds of kinetic energy at once
- A rolling body has translational KE $\tfrac12 mv^2$ and rotational KE $\tfrac12 I\omega^2$.
- Its total kinetic energy is the sum: $E_k = \tfrac12 mv^2 + \tfrac12 I\omega^2$.
- Rolling without slipping links the two through $v = r\omega$.
- Energy released (say from a ramp) must fill both accounts.
The total kinetic energy of a rolling object is:
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A rolling body moves and spins, so its KE is the sum $\tfrac12 mv^2 + \tfrac12 I\omega^2$.
Rolling without slipping links linear and angular speed by $v = r\_\_$ (fill in the symbol).
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$v = r\omega$ ties how fast it travels to how fast it spins.
Why the solid disk wins
- The ring has more rotational inertia, so more of its energy goes into spinning.
- That leaves less energy for moving, so the ring accelerates down the ramp more slowly.
- The solid disk keeps more energy for translation and reaches the bottom first.
- Shape decides the race, not weight — a heavy and light disk of the same shape tie.
A solid disk and a hollow ring of the same mass and radius roll down a ramp. Which reaches the bottom first?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The ring has more rotational inertia, so more energy goes to spinning, leaving less for moving — the disk wins.
Select all true statements about a body rolling without slipping.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Rolling combines moving and spinning, obeys $v = r\omega$, and has a momentarily still contact point — but not all rotational.
The contact point is momentarily still
- In rolling without slipping, the point touching the ground is instantaneously at rest.
- The centre moves at $v$, and the top of the wheel moves at $2v$.
- Because the contact point doesn't slide, static friction acts — and does no work.
- This is why a rolling wheel loses so little energy compared with a sliding block.
Rolling without slipping
For a wheel that rolls without slipping, v = r omega. Sort each case.
For a wheel rolling without slipping, the point touching the ground is momentarily at rest.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The contact point has zero velocity at that instant; the top moves at $2v$.
A rolling object's energy is not just $\tfrac12 mv^2$. You must add the rotational part $\tfrac12 I\omega^2$. Forgetting the spin term gives too large a speed at the bottom of a ramp — a very common mistake.
A solid disk ($I = \tfrac12 mR^2$) rolls without slipping. What fraction of its kinetic energy is rotational? (Give a decimal.)
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Rotational $= \tfrac14 mv^2$, translational $= \tfrac12 mv^2$; fraction $= \tfrac{1/4}{3/4} = \tfrac13 \approx 0.33$.
A solid disk ($I = \tfrac12 mR^2$) rolls without slipping at speed $v$. What fraction of its kinetic energy is rotational?
- Rotational: $\tfrac12 I\omega^2 = \tfrac12(\tfrac12 mR^2)(v/R)^2 = \tfrac14 mv^2$.
- Translational: $\tfrac12 mv^2$. So rotational is $\tfrac{1/4}{1/4 + 1/2} = \tfrac13$ of the total.
A rolling body has both translational KE $\tfrac12 mv^2$ and rotational KE $\tfrac12 I\omega^2$, linked by $v = r\omega$. Objects with more rotational inertia put more energy into spinning, so they roll down a ramp slower. The contact point is momentarily at rest, so rolling friction does no work.
Write from a ring whose spin holds half the energy
- A thin ring has rotational inertia m R squared and rolls without slipping.
- Find the fraction of its kinetic energy that is rotational.
A thin ring with rotational inertia equal to m R squared rolls without slipping. What fraction of its kinetic energy is rotational?
Two equal shares of one half m v squared make the rotational fraction one half.
What fraction of the ring's kinetic energy is rotational?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
For a thin ring the spin and the forward motion store equal kinetic energy. Other options — About 0.33 (the solid-disk fraction); All of it (because a ring only spins).