1) What it calculates
An angular to linear acceleration calculator turns a rotational rate of change (angular acceleration) into a straight-line rate of change (linear acceleration) for a point on a rotating object. It is used for wheels, pulleys, gears, spindles, turntables, and robotic joints—anywhere rotational motion produces measurable acceleration at a distance from an axis.
The output is usually the tangential acceleration at a chosen radius. That value tells you how fast the speed of the point increases along its circular path, expressed in linear units such as m/s² or ft/s².
2) The core idea
In rigid-body rotation, every point shares the same angular acceleration, but points farther from the center travel a longer arc for the same angular change. That geometry is why the rim of a wheel experiences more linear acceleration than a point near the hub.
Think of radius as a “multiplier” that converts rotational change into linear change. Double the radius and, for the same angular acceleration, you double the tangential acceleration.
3) Formula used
The calculator is built around a direct relationship between angular acceleration and tangential acceleration:
at = α × r
- at: tangential (linear) acceleration
- α: angular acceleration
- r: radius to the point of interest
This result applies to a specific point on the rotating body. If you need acceleration at another location, use that location’s radius.
4) Choosing the right radius
Selecting the radius is where real-world setups differ. A “wheel radius” might mean the rim radius, the effective rolling radius at the tire contact patch, or the pitch radius of a sprocket. Likewise, a gear “radius” usually means the pitch circle radius rather than the outside diameter.
Use a radius that matches the problem you are solving:
- Traction and vehicle acceleration: use effective rolling radius (loaded tire radius), not the free tire radius.
- Belt speed and belt acceleration: use pulley pitch radius where the belt sits.
- Stress at an edge: use the actual distance to the edge that experiences the load.
- Robot arm end point: use the distance from the joint axis to the tool tip.
If your system has slip (tire slip, belt slip) or compliance (flexible shafts), treat the calculated value as an ideal acceleration at the rotating part.
5) Units and conversions
Most calculators accept α in rad/s² and radius in m, producing m/s². Problems usually happen when units are mixed.
Degrees vs. radians
The formula assumes radians. If your α is in degrees/s², convert it:
αrad/s² = αdeg/s² × (π/180).
Diameter vs. radius
If you have diameter D, use r = D/2. Entering diameter where radius is expected doubles the calculated acceleration.
Length units
Keep radius in one unit system (m, cm, ft, in). The output follows that system. For example, inches with rad/s² yields in/s².
6) How to use the calculator
- Pick the point you care about. Rim of a wheel, pitch radius of a gear, belt contact point, or tool tip on a rotating arm.
- Enter the radius. Use distance from the axis to that point; convert diameter to radius if needed.
- Enter angular acceleration. Confirm it is in rad/s² (convert from degrees/s² if necessary).
- Read tangential acceleration. Interpret it as the linear speed change per second at that radius.
- Optional: compare to limits. Check traction, belt tension, structural loads, or comfort constraints.
7) Worked examples
Example A: Wheel rim
Given α = 12 rad/s² and r = 0.35 m:
at = 12 × 0.35 = 4.2 m/s².
So the rim speed increases by 4.2 m/s each second (tangentially).
Example B: Convert degrees/s²
Given α = 300 deg/s² and r = 0.20 m. Convert:
α = 300 × (π/180) ≈ 5.236 rad/s².
Then:
at ≈ 5.236 × 0.20 = 1.047 m/s².
Example C: Diameter entered by mistake
A pulley has D = 0.50 m and α = 8 rad/s². Correct radius is 0.25 m:
at = 8 × 0.25 = 2.0 m/s².
Reference values for α = 10 rad/s²:
| Radius r | at = αr | Interpretation |
|---|---|---|
| 0.10 m | 1.0 m/s² | Gentle speed increase |
| 0.50 m | 5.0 m/s² | Noticeably brisk |
| 1.00 m | 10.0 m/s² | Near 1 g scale |
8) Tangential vs. centripetal acceleration
A rotating point can have two accelerations. Tangential comes from changing speed (angular acceleration) and is what this conversion gives:
at = αr.
Centripetal comes from changing direction and exists when angular velocity is nonzero:
ac = ω2r.
Even if α = 0, a fast-spinning wheel can still generate large centripetal acceleration.
9) Typical use cases
- Robotics: convert joint angular acceleration to tool-tip tangential acceleration using link length.
- Vehicles: relate wheel angular acceleration to forward acceleration at the tread (ignoring slip).
- Belts and pulleys: estimate belt acceleration for tensioning, tracking, and startup analysis.
- Machinery: translate spindle dynamics into surface acceleration at the workpiece radius.
- Education: connect rotational kinematics to familiar linear motion concepts.
10) Common mistakes
- Degrees used as radians: always convert deg/s² to rad/s².
- Diameter used as radius: divide diameter by two.
- Mixed length units: decide on meters (or feet) and stick with it.
- Wrong radius location: state the exact point (rim, pitch radius, tip) your result represents.
- Confusing α and ω: α changes speed; ω is the current spin rate.
11) Quick result checks
- Scaling: doubling r should double at.
- Zero test: if α = 0, then at must be 0.
- Magnitude: compare to gravity (~9.81 m/s²). Values much larger can be demanding for traction and structural limits.
- Radian conversion: if your input was in degrees, confirm the result is smaller by about 57.3×.
12) Summary
An angular to linear acceleration calculator uses at=αr to convert rotational acceleration into tangential acceleration at a chosen
radius. For reliable results, use radians, enter radius (not diameter), keep length units consistent, and be clear about the point you are analyzing. When loads
matter, consider centripetal acceleration as well. When sharing results, state the radius and units so others can reproduce them without hidden conversion errors.