Rolling Motion
- Apply the rolling-without-slipping condition, v = rω
- Explain why different shapes reach the bottom of a ramp at different speeds
- Connect rolling to combined translation and rotation
Rolling without slipping links v and ω
Rolling without slipping is the condition where a wheel’s contact point does not skid. It ties the forward speed of the center to the spin rate: v = rω. In one full turn the wheel advances exactly one circumference. Remarkably, the contact point is instantaneously at rest against the ground, which is why static (not kinetic) friction acts and no energy is dissipated.
The ramp race depends on shape
Release several round objects from the same height and they arrive with different speeds, because gravitational PE splits between translation (½mv²) and rotation (½Iω²). Shapes with more rotational inertia relative to mR² — like a hoop — divert more energy into spin, leaving less for forward motion, so they arrive slower. A solid sphere, with mass concentrated near its axis, wins the race.
A wheel of radius 0.3 m rolls without slipping so its center moves at 6 m/s. Find its angular speed.
- 1.Use the rolling condition v = rω.
- 2.Rearrange for ω: ω = v ÷ r.
- 3.Substitute: ω = 6 ÷ 0.3.
- 4.Compute: ω = 20 rad/s.
Rolling without slipping means static friction, not kinetic — so it does no work and mechanical energy is conserved. Only when a wheel skids does friction dissipate energy as heat.
A wheel of radius 0.5 m rolls without slipping while its center moves at 10 m/s. What is its angular speed?
A solid sphere and a hollow hoop of the same mass and radius are released from rest at the top of the same ramp. Which reaches the bottom first?
For ramp-race questions, rank shapes by how much of mgh goes into rotation: sphere (⅖MR²) beats disk (½MR²) beats hoop (MR²). Mass and radius cancel out, so the shape alone decides the winner.
Answer the 2 checkpoints as you read.
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