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Mechanical

Radial acceleration in polar coordinates

Radial component of acceleration for a point in planar polar coordinates.

Ref: MEC-12
Rev: v1
Field: Mechanical
Result: a_r [m/s²]

a_{r} = r̈ - r {θ̇}^{2}

Fixed origin, outward radial direction. r̈ is the second derivative of r with time and θ̇ the angular rate in rad/s. The −rθ̇² term is the centripetal contribution (it points inward).

A point on a rotating arm accelerates inward because of the rotation, whatever it does radially.

Calculator

Solve for

a_rRadial accelerationComputed

m/s²

rRadius

0.120

r̈Radial acceleration of r

m/s²

−1010

θ̇Angular rate

rad/s

0.120

Centripetal term r θ̇²18 m/s²
Rim speed r θ̇6 m/s

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Radial acceleration (m/s²) current value −17.8 m/s²

θ̇Angular rateKey driver

−8000.18

rRadius

−180−0.7

r̈Radial acceleration of r

−28−8

Even with r constant (r̈ = 0) the radial acceleration is −rθ̇², the usual centripetal result.
The negative sign shows the acceleration points toward the origin.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m/s²)	Local step	Change in result
θ̇ Angular rate	0.1 to 20 rad/s	−799.8	0.18	800	+10%	+21.2 %
r Radius	0.1 to 20 m	−179.8	−0.7	179.1	+10%	+10.1 %
r̈ Radial acceleration of r	−10 to 10 m/s²	−28	−8	20	+10%	−0.112 %

Sweep of Angular rate

Angular rate (rad/s)	Radial acceleration (m/s²)
0.1	0.18
2.587	−13.19
5.075	−51.31
7.562	−114.2
10.05	−201.8
12.54	−314.2
15.02	−451.3
17.51	−613.2
20	−799.8

Sweep one variable

Vary

Radial acceleration (m/s²)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

a_r

Radial acceleration

Result · m/s²

What the equation gives you. Everything else feeds it.

Radius

m · typical 0.1 to 20

Distance from the origin. The rotation term scales with r.

r̈

Radial acceleration of r

m/s² · typical −10 to 10

Second time derivative of the radius. Adds directly.

θ̇

Angular rate

rad/s · typical 0.1 to 20

Angular velocity of the radius line. Squared: the strongest driver at high rotation rates.

Worked example with the values above

Centripetal term: rθ̇² = 2 × 3² = 18 m/s².
Radial term: r̈ = 0.2 m/s².
a_r = 0.2 − 18 = −17.8 m/s² (negative means inward).

Assumptions and limits

Fixed origin and outward radial axis.
θ̇ in radians per second.

Common mistakes

Forgetting the −rθ̇² term.
Using rpm directly for θ̇.

Statistics and uncertainty

Rotation rate is squared, so a 2% error in θ̇ gives a 4% error in the centripetal term.

Where it is used

Rotating arms, radar and gun mounts.
Slider-on-rotating-rod problems.

Maths you need

Polar coordinates: r and θ
Derivatives: first and second time derivatives

Related formulas

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