Rolling
Introduced| English | Meaning |
|---|---|
| rolling without slipping/ˈrəʊlɪŋ wɪˈðaʊt ˈslɪpɪŋ/ | motion in which a wheel or ball turns so that its contact point is momentarily at rest, so v = rω |
Why a ball beats a ring downhill
- Race a solid ball and a hollow ring down the same ramp.
- Same mass, same radius, same start — yet the ball wins every time.
- The secret is how each one shares its motion.
- A wheel that turns as it moves obeys a neat speed rule.
Rolling without slipping
- When a wheel rolls without slipping, its speed and spin lock together:
- The contact point is momentarily at rest on the ground.
- Speeding up obeys the matching rule $a = \alpha r$.
A wheel of radius $0.3\ \text{m}$ rolls without slipping at $\omega = 10\ \text{rad/s}$. Its forward speed (in m/s)?
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$v = \omega r = 10 \times 0.3 = 3\ \text{m/s}$.
For a wheel rolling without slipping, the point touching the ground is momentarily...
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The contact point has zero velocity at that instant -- that is the no-slip condition.
While a wheel of radius r rolls without slipping, its linear acceleration a equals alpha times ____.
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$a = \alpha r$ -- the acceleration version of $v = \omega r$.
Motion shared two ways
- A rolling object's kinetic energy splits into two parts:
- Part goes into moving forward, part into spinning.
- Using $v = \omega r$, both parts grow together.
A rolling object's kinetic energy is split between translation and rotation.
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$K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$ -- moving and spinning.
Who reaches the bottom first
- On a ramp, the released energy splits between forward motion and spin.
- A shape with less rotational inertia (per $mr^2$) keeps more for speed.
- Order: solid sphere, then solid cylinder, then hollow ring — the sphere wins.
Rolling without slipping
For a wheel that rolls without slipping, v = r omega. Sort each case.
Order these from FIRST to LAST reaching the bottom of the same ramp (same m, r).
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Smaller rotational inertia keeps more energy for speed, so the sphere leads and the ring trails.
A hoop and a solid disc, same $m$ and $r$, roll from rest down the same ramp.
- Hoop: $I = mr^2$, so half its energy goes to spin — it is slow.
- Disc: $I = \tfrac{1}{2}mr^2$, a smaller spin share — it arrives first.
During rolling without slipping, how much work does the static friction do?
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No sliding at the contact means no friction work and no heat -- unlike a skidding wheel.
Rolling without slipping needs static friction at the contact, but that friction does no work — the contact point does not slide, so there is no friction-heat loss. Only a skidding (slipping) wheel loses energy to kinetic friction. And $v = \omega r$ links speed to spin only while it rolls without slipping.
Rolling without slipping 无滑滚动 locks speed to spin: $v = \omega r$ (and $a = \alpha r$). The kinetic energy splits into $\tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$, so a shape with smaller rotational inertia wins a downhill race (sphere before cylinder before ring). Static friction enables the roll but does no work.
Write from a forward speed of 2 m/s
- A wheel of radius 0.40 m rolls without slipping at 5 rad/s.
- Find the forward speed.
A wheel of radius 0.40 m rolls without slipping at 5 rad/s. What is its forward speed? What is the speed of the point touching the ground?
5 times 0.40 is 2.
What is the forward speed?
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Angular speed times radius is 5 times 0.40. Other options — 3 m/s (the lesson rolling speed); 5 m/s (the angular speed used as a linear speed).