What is the angular acceleration?
Choose what you know, then type the speeds and time, the angle turned, the torque and moment of inertia, or the tangential acceleration and radius. The angular acceleration calculator gives α in rad/s², °/s², rev/s² and rpm per second.
- Answer
- α = 5.23599 rad/s² (300°/s²)
The angular acceleration is 5.23599 rad/s².
- Angular acceleration (rad/s²)
- 5.23599
- Angular acceleration (°/s²)
- 300
- Angular acceleration (rev/s²)
- 0.833333
- Angular acceleration (rpm/s)
- 50
- Angle turned (rev)
- 10.4167
- Angle turned (rad)
- 65.4498
Answer: α = 5.23599 rad/s² (300°/s²). The angular acceleration is 5.23599 rad/s².
How to calculate
Angular acceleration calculator: α = Δω ÷ Δt from two angular speeds and a time, or from the angle turned, torque and moment of inertia (α = τ ÷ I), or tangential acceleration (α = a ÷ r).
Example with the default inputs (From Speeds and time, Starting angular speed (ω₀) 0, Final angular speed (ω) 250, Angular speed unit rpm (rev/min), Time (Δt) 5 s): The angular acceleration is 5.23599 rad/s².
Method: α = (ω − ω₀) ÷ Δt; α = (ω² − ω₀²) ÷ (2Δθ) with Δt = 2Δθ ÷ (ω₀ + ω); α = τ ÷ I; α = a_t ÷ r; 1 rpm = 2π ÷ 60 rad/s.
- The angular acceleration is constant over the time or angle (the average for a change that is not steady).
- A rigid body turning about a fixed axis; the torque and the moment of inertia are about that same axis.
Worked examples
Each example is checked against the calculator on every build.
- From Speeds and time, Starting angular speed (ω₀) 0, Final angular speed (ω) 250, Angular speed unit rpm (rev/min), Time (Δt) 5 s gives Angular acceleration (rad/s²) 5.235988, Angular acceleration (°/s²) 300, Angular acceleration (rpm/s) 50, Angle turned (rev) 10.416667, Answer α = 5.23599 rad/s² (300°/s²).Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19)
- From Speeds and time, Starting angular speed (ω₀) 26.2, Final angular speed (ω) 0, Angular speed unit rad/s, Time (Δt) 0.3 s gives Angular acceleration (rad/s²) -87.333333, Angular acceleration (°/s²) -5,003.831411.Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Example 10.2(b) (α = −87.3 rad/s² stops 26.2 rad/s in 0.300 s)
- From Speeds and angle, Starting angular speed (ω₀) 0, Final angular speed (ω) 220, Angular speed unit rad/s, Angle turned (Δθ) 220, Angle unit rad gives Angular acceleration (rad/s²) 110, Time taken (s) 2.Source: OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration (Equations 10.11 to 10.13; Example 10.4: a reel at 110 rad/s² reaches 220 rad/s after 220 rad and 2.00 s), https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration (retrieved 2026-10-19)
- From Torque and inertia, Net torque (τ) 375, Torque unit N·m, Moment of inertia (I) 56.25, Inertia unit kg·m² gives Angular acceleration (rad/s²) 6.666667.Source: OpenStax, University Physics Volume 1, §10.7 Newton’s Second Law for Rotation (Equation 10.25, Στ = Iα; Example 10.16: 375 N·m on 56.25 kg·m² gives 6.67 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation (retrieved 2026-10-19)
- From Tangential acceleration, Tangential acceleration (a_t) 4.95, Acceleration unit m/s², Radius (r) 1.772 in gives Angular acceleration (rad/s²) 110.Source: OpenStax, University Physics Volume 1, §10.1 Rotational Variables (Equations 10.6 and 10.8; Example 10.2: 0 to 250 rpm in 5.00 s is 5.24 rad/s²), https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables (retrieved 2026-10-19), Equation 10.8 (a_t = rα), with OpenStax, University Physics Volume 1, §10.2 Rotation with Constant Angular Acceleration (Equations 10.11 to 10.13; Example 10.4: a reel at 110 rad/s² reaches 220 rad/s after 220 rad and 2.00 s), https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration (retrieved 2026-10-19): 4.95 ÷ 0.045 = 110 rad/s²
How it works
Pick what you know:
- Speeds and time: α = (ω − ω₀) ÷ Δt. The page also gives the angle turned, Δθ = (ω₀ + ω) ÷ 2 × Δt, in turns and radians.
- Speeds and angle: from ω² = ω₀² + 2αΔθ, α = (ω² − ω₀²) ÷ (2Δθ). The page also gives the time, Δt = 2Δθ ÷ (ω₀ + ω).
- Torque and inertia: α = τ ÷ I (Newton's second law for rotation).
- Tangential acceleration: α = a_t ÷ r.
The answer shows in rad/s², °/s², rev/s² and rpm per second. A negative α slows a spin that is going forward.
Units (exact): 1 rpm = 2π ÷ 60 rad/s; 1°/s = π ÷ 180 rad/s; 1 rev/s = 2π rad/s; 1 rev = 2π rad; 1° = π ÷ 180 rad; 1 lbf·ft = 4.4482216152605 × 0.3048 N·m; 1 lbf·in = 4.4482216152605 × 0.0254 N·m; 1 g·cm² = 10⁻⁷ kg·m²; 1 lb·ft² = 0.45359237 × 0.3048² kg·m²; 1 lb·in² = 0.45359237 × 0.0254² kg·m²; 1 cm/s² = 0.01 m/s²; 1 ft/s² = 0.3048 m/s²; 1 g = 9.80665 m/s²; 1 ms = 0.001 s; 1 min = 60 s; 1 h = 3,600 s; lengths mm, cm, m, in (0.0254 m) and ft (0.3048 m).
Rules
- Angular speeds are numbers from −10⁹ to 10⁹ in the unit chosen (both speeds share it); time from 10⁻⁹ s to 10⁹ s; angle turned from 10⁻⁹ to 10¹² in its unit; torque from −10¹² to 10¹²; moment of inertia from 10⁻¹² to 10¹⁵; tangential acceleration from −10⁹ to 10⁹ in their units; radius from 10⁻⁹ m to 10⁹ m. Limits include their ends; a time or radius typed in another unit is converted first.
- With speeds and an angle, ω₀ + ω must be more than 0: a positive angle needs a forward spin. Otherwise there is no answer.
- A result that is exactly 0 shows as 0 (equal speeds give α = 0). Any other result too large or too small to show as a number gives no answer.
Output format
The answer line reads α = a rad/s² (b°/s²). It and every number show to 6 significant figures, rounded half up, with trailing zeros dropped and thousands commas, and below 10⁻⁶ or from 10¹⁵ up as a × 10ⁿ. Negative numbers show with a minus sign (−). Typed values are read back as the exact decimals you typed, and every unit size is an exact fraction times a power of π (1 rpm = π ÷ 30 rad/s), so a result is an exact fraction until it is multiplied by π once (or not at all, as in rad/s from rad/s).
Worked examples by hand
A wheel spun up. OpenStax Example 10.2: 0 to 250 rpm in 5.00 s. 250 rpm = 250 × 2π ÷ 60 = 26.1799 rad/s, so α = 26.1799 ÷ 5 = 5.23599 rad/s² (= 50 rpm/s = 300°/s²). It turns (0 + 250) ÷ 2 × 5 ÷ 60 = 10.4167 rev on the way.
Braking. From 26.2 rad/s to 0 in 0.300 s: α = (0 − 26.2) ÷ 0.3 = −87.3333 rad/s², which is −5,003.83°/s².
From the angle turned. OpenStax Example 10.4: a fishing reel goes from rest to 220 rad/s while it turns 220 rad. α = (220² − 0²) ÷ (2 × 220) = 110 rad/s², and Δt = 2 × 220 ÷ (0 + 220) = 2 s.
From torque. OpenStax Example 10.16: a 250 N push at the edge of a 1.50 m merry-go-round gives τ = 375 N·m, with I = 56.25 kg·m²: α = 375 ÷ 56.25 = 6.66667 rad/s².
From tangential acceleration. The reel's line comes off a 4.50 cm radius at a_t = 4.95 m/s²: α = 4.95 ÷ 0.045 = 110 rad/s².
Other questions people ask
How do you calculate angular acceleration?
Divide the change in angular speed by the time it takes: α = (ω − ω₀) ÷ Δt. A wheel spun from rest to 250 rpm (26.18 rad/s) in 5.00 s has α = 26.18 ÷ 5 = 5.24 rad/s².
How do you convert rpm to rad/s?
Multiply by 2π and divide by 60: one turn is 2π radians and one minute is 60 seconds. 250 rpm = 250 × 2π ÷ 60 = 26.18 rad/s. In the same way, an angular acceleration of 1 rpm per second is 0.10472 rad/s².
How do you find angular acceleration from torque?
Use Newton’s second law for rotation, α = τ ÷ I. A 375 N·m torque on a merry-go-round with I = 56.25 kg·m² gives α = 375 ÷ 56.25 = 6.67 rad/s².
How do you find angular acceleration without the time?
Use the angle turned: ω² = ω₀² + 2αΔθ, so α = (ω² − ω₀²) ÷ (2Δθ). A reel that reaches 220 rad/s from rest within 220 rad has α = 220² ÷ (2 × 220) = 110 rad/s², and the time is 2Δθ ÷ (ω₀ + ω) = 2 s.
What is the relation between angular and tangential acceleration?
A point at radius r speeds up along its path at a_t = rα, so α = a_t ÷ r. The rim of a 4.5 cm reel speeding up at 4.95 m/s² gives α = 4.95 ÷ 0.045 = 110 rad/s².
What does a negative angular acceleration mean?
The spin is slowing down (or speeding up in the other direction). A wheel braked from 26.2 rad/s to rest in 0.300 s has α = −26.2 ÷ 0.3 = −87.3 rad/s².