 # Angular Acceleration

## Angular Acceleration Definition

It is the rate of change of angular velocity with a time of an object in motion. Acceleration is the change in velocity of a moving object with respect to time. If the object moves on a circular direction than its velocity is called angular velocity. The angular acceleration is also known as rotational acceleration. It is a quantitative expression of the change in angular velocity per unit time. The acceleration vector, magnitude, or length is directly proportional to the rate of change in angular velocity.

## Units of Angular Acceleration

The vector direction of the acceleration is perpendicular to the plane where the rotation takes place. Increase in angular velocity clockwise, then the angular acceleration velocity points away from the observer. If the increase in angular velocity counterclockwise, then the vector of angular acceleration points toward the viewer.

The amount of angle covered per square of time is measured in radians per second squared. The unit is degrees/square of time like s2, min2, hr2.

In SI units, it is measured in radians per second squared (rad/s2) and is usually denoted by the alpha (α).

The angular acceleration can also be defined as either:

 $\alpha =\frac{d\omega }{dt}=\frac{\mathrm{d}^{2} \theta }{\mathrm{d} t^{2}}$Or$\alpha =\frac{a_{T}}{r}$

Where,

• ω = Angular Velocity
• aT = Linear Tangential Acceleration
• r = Radius of Circular Path

## Angular Acceleration Formula

The rate of change of angular velocity regarding time is angular acceleration, which is a vector quantity. It is denoted by α. The angular acceleration formula is given by,

 $\omega =\frac{\theta }{t}$

Where,

• ω = Angular Velocity
• θ = Angle Rotated
• t = Time Taken

When the angular velocity is constant, the angular acceleration is 0. If the velocity is not constant, then the constant α is defined as:

 $\alpha =\frac{\omega }{t}=\frac{\theta }{t^{2}}$

If the angular acceleration is not constant and differs from time to time, then:

 $\alpha_{avg}=(\frac{\omega_{2} -\omega _{1}}{t_{2}-t_{1}})$

And,

 $\alpha_{i}=\frac{d\omega }{dt}=\frac{d^{2}\theta }{dt^{2}}$

The Earth takes 23 hours and 56 minutes to complete one revolution and rotates at an angular velocity of 7.2921159 × 10-5radians/second.

### Average Angular Velocity Formula

Average angular velocity is the ratio of change in angular coordinate to change in time. Following is the formula of average angular velocity:

 $\omega _{avg}=\frac{\Delta \theta }{\Delta t}=\frac{\theta _{2}-\theta _{1}}{t_{2}-t_{1}}$

Where,

• ωavg is the average angular velocity in radian.s-1
• ΔӨis the change in angular coordinate in radians
• Ө1 is the initial angular coordinate in radians
• Ө2 is the final angular coordinate in radians
• Δt is the change in time in s
• t1 is the initial time in s
• t2 is the final time in s

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