Angle Traced in Nth Second (Accelerated Rotatory Motion) Calculator

Formula

The formula to calculate the Angular Displacement (θ) in the Nth Second of Accelerated Rotatory Motion is:

\[ θ = ωo + \left(\frac{2n - 1}{2}\right) \cdot α \]

Where:

\(θ\) is the Angular Displacement (measured in radians).
\(ωo\) is the Initial Angular Velocity (measured in radians per second).
\(n\) is the Nth Second (measured in seconds).
\(α\) is the Angular Acceleration (measured in radians per square second).

Definition

Angular Displacement is defined as the shortest angle between the initial and the final points for a given object undergoing circular motion about a fixed point.

Initial Angular Velocity is the velocity at which motion starts.

The Nth Second is the n seconds time covered by the body.

Angular Acceleration refers to the time rate of change of angular velocity.

How to calculate Angular Displacement in Nth Second

Let's assume the following values:

Initial Angular Velocity (ωo) = 14 rad/s
Nth Second (n) = 4 seconds
Angular Acceleration (α) = 1.6 rad/s²

Using the formula:

\[ θ = 14 + \left(\frac{2 \cdot 4 - 1}{2}\right) \cdot 1.6 \]

Evaluating:

\[ θ = 14 + \left(\frac{7}{2}\right) \cdot 1.6 = 14 + 3.5 \cdot 1.6 = 14 + 5.6 = 19.6 \]

The Angular Displacement is 19.6 radians.

Conversion Chart

Initial Angular Velocity (ωo) (rad/s)	Nth Second (n) (s)	Angular Acceleration (α) (rad/s²)	Angular Displacement (θ) (rad)
10	2	1.2	11.8000
10	2	1.6	12.4000
10	2	2	13.0000
10	4	1.2	14.2000
10	4	1.6	15.6000
10	4	2	17.0000
10	6	1.2	16.6000
10	6	1.6	18.8000
10	6	2	21.0000
14	2	1.2	15.8000
14	2	1.6	16.4000
14	2	2	17.0000
14	4	1.2	18.2000
14	4	1.6	19.6000
14	4	2	21.0000
14	6	1.2	20.6000
14	6	1.6	22.8000
14	6	2	25.0000
18	2	1.2	19.8000
18	2	1.6	20.4000
18	2	2	21.0000
18	4	1.2	22.2000
18	4	1.6	23.6000
18	4	2	25.0000
18	6	1.2	24.6000
18	6	1.6	26.8000
18	6	2	29.0000