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Rolling Motion Calculator

Acceleration of a rolling object on an incline without slipping (a = g sin θ / (1 + c) where c = I/(mr²))

Parameters

°ⓘ
ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Rolling Acceleration:
3.50;m/s23.50;m/s²
Sliding Acceleration (no roll):
4.90;m/s24.90;m/s²
Fraction of g sin θ:
0.71;0.71;
Inertia Factor c:
0.40;0.40;

Examples

Sphere at 30°

Solid sphere on 30° incline.

  • Rolling Acceleration: 3.503.50

Visualization

Rolling Without Slipping

Rolling combines translation and rotation. Without slipping, the contact point has zero velocity and v = ωr links linear and angular speed.

On an incline, gravity does work that becomes both translational KE (½mv²) and rotational KE (½Iω²). With I = cmr², acceleration is a = g sin θ/(1 + c).

Solid sphere: c = 2/5 → a = (5/7)g sin θ. Solid cylinder: c = 1/2 → a = (2/3)g sin θ. Hollow cylinder: c = 1 → a = ½g sin θ.

Rolling is slower than sliding (same θ) because energy is shared with rotation. Static friction enables rolling without dissipating work (ideal model).

If slipping occurs, v ≠ ωr and the simple energy derivation fails; use dynamics with friction equations.

Key Concepts

  • v = ωr — no-slip condition
  • a = g sin θ/(1 + c) — rolling on incline
  • c = I/(mr²) — dimensionless inertia factor
  • KE_total = ½mv²(1 + c) at bottom
  • Static friction, no slip: no work at contact
  • Lower c → faster roll (sphere fastest)

Real-World Applications

  • Balls and cylinders on ramps (lab demos)
  • Wheels, yo-yos, and rolling vehicles
  • Class 11 rotational mechanics chapter
  • Engineering: rollers and bearings
  • Comparing slide vs roll race experiments

Explore Further

More mechanics tools

Physics Equations

Rolling Acceleration:
a=gsin⁡θ1+ca = \frac{g \sin\theta}{1 + c}
No-Slip Condition:
v=ωrv = \omega r

Step-by-Step Solution

See how the main results are calculated.

1

Set Up Energy Conservation

Rolling without slipping: v = ωr. At top, KE = 0; at distance s down the incline:

Equation:

mgh=12mv2+12Iω2=12mv2(1+c)mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 = \frac{1}{2}mv^2(1 + c)

Calculation:

I=cmr2,c=0.4 for Solid SphereI = cmr^2, \quad c = 0.4 \text{ for Solid Sphere}

Result:

Gravitational PE converts to translational + rotational KE

Explanation:

Static friction causes rotation but does no work (contact point has zero displacement). No slipping is essential.

2

Derive Acceleration

From Newton’s laws or energy, for rolling down incline angle θ:

Equation:

a=gsin⁡θ1+ca = \frac{g\sin\theta}{1 + c}

Calculation:

sin⁡(30°)=0.5000\sin(30°) = 0.5000
a=9.81×0.50001+0.4=3.50 m/s2a = \frac{9.81 \times 0.5000}{1 + 0.4} = 3.50 \text{ m/s}^2

Result:

a=3.50 m/s2a = 3.50 \text{ m/s}^2

Explanation:

Solid Sphere: smaller c → more translational KE → faster descent. Sliding (no rotation): a = g sin θ = 4.90 m/s².

3

Compare to Sliding

Fraction of sliding acceleration achieved when rolling:

Equation:

arollaslide=11+c\frac{a_{roll}}{a_{slide}} = \frac{1}{1 + c}

Calculation:

3.504.90=0.714\frac{3.50}{4.90} = 0.714

Result:

Rolling is 29% slower than sliding

Explanation:

Hollow cylinder (c = 1) has the slowest roll; solid sphere (c = 2/5) is fastest among common shapes.

Frequently Asked Questions (FAQ)

Why does a sphere roll faster than a hollow cylinder?

Lower inertia factor c means less energy goes into rotation, so more goes into translation—greater a.

What is rolling without slipping?

The contact point has zero velocity; v = ωr links linear and angular motion.

Does friction do work in ideal rolling?

Static friction at the contact point does no work when there is no slipping.

When does the simple formula fail?

When slipping occurs (v ≠ ωr) or when friction is insufficient to maintain roll.

Practice MCQs

  1. Compared to sliding, rolling acceleration is:
  2. Solid sphere has inertia factor c =
  3. No-slip condition:

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