Rolling
Introduced| English |
|---|
| rolling without slipping/ˈrəʊlɪŋ wɪˈðaʊt ˈslɪpɪŋ/ |
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)?
$v = \omega r = 10 \times 0.3 = 3\ \text{m/s}$.
For a wheel rolling without slipping, the point touching the ground is momentarily...
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 ____.
$a = \alpha r$ -- 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.
$K = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$ -- 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).
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?
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.