Rolling Down an Incline Calculator (Sphere, Cylinder, Hoop Race)
Acceleration, speed at the bottom and time for a sphere, cylinder or hoop rolling down a slope, and the friction needed so it does not slip.
Results
What this tool does
Roll a solid sphere, a hollow ball, a can and a ring down the same slope and the finishing order never changes — solid sphere first, then the solid cylinder, the hollow sphere, and the ring last — whatever their masses and sizes. The reason is that some of the energy goes into spin: a = g sin θ ÷ (1 + k), where k = I ÷ (m r²) measures how far from the axis the mass sits. A block sliding without friction (k = 0) beats them all. Rolling without slipping needs enough friction, at least tan θ × k ÷ (1 + k); on ice or an oily slope the objects slide instead and the formula no longer applies.
Formula
a = g sin θ ÷ (1 + k), k = I ÷ (m r²) · v = √(2 a L) · t = √(2L ÷ a) · μ ≥ tan θ × k ÷ (1 + k)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
sh | Shape of the rolling object | — |
an | Angle | ° |
sl | Length of the slope (distance along it) | m |
A | Acceleration down the slope | m/s² |
V | Speed at the bottom (centre of mass) | m/s |
T | Time to reach the bottom | s |
RS | Share of the energy that goes into spinning | % |
MU | Least friction coefficient to roll without slipping | — |
Worked example
- Shape of the rolling object1
- Angle20 °
- Length of the slope (distance along it)3 m
- Acceleration down the slope2.396 m/s²
- Speed at the bottom (centre of mass)3.791 m/s
- Time to reach the bottom1.583 s
- Share of the energy that goes into spinning28.6 %
- Least friction coefficient to roll without slipping0.104
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.