Angular Acceleration 

Angular acceleration is referred to as the rate of change of angular velocity with respect to time (t). It is a vector quantity measured in rad/s². It plays a key role in rotational motion and dynamics.

1.0Insight into Angular Acceleration 

Angular acceleration is basically the rate of change of the angular velocity of an object with respect to time. It is denoted by 𝞪 with the unit of measurement rad/s². It is a vector quantity, meaning it has both magnitude and direction. It can be mathematically represented as: 

$\alpha =\frac{d\omega }{dt}$

Key Concepts

• Angular Displacement (θ): It is the angle through which an object rotates along a particular axis with a specific angle θ.

• Angular Velocity ($\omega$) = It is defined as the rate of change of angular displacement with respect to time. It is measured in radians per second (rad/s). $\omega =\frac{d\theta }{dt}$

2.0Angular Acceleration Formula 

When the angular velocity varies uniformly with time, then the angular acceleration is constant and can be calculated by using these equations:

1. First Equation of Rotational Motion: 

$\omega ={\omega }_0+\alpha t$

Where:

• ω₀​ = angular velocity.
• ω = final angular velocity after time t.
• α = angular acceleration.
• t = time elapsed.

2. Second Equation of Rotational Motion:

$\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$

Where:

• θ₀ = initial angular displacement.
• θ = final angular displacement after time t.
• α = angular acceleration.
• t = time.

3. Third Equation of Rotational Motion:

${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$

Where:

• ω0 = initial angular velocity.
• ω = final angular velocity.
• θ−θ₀ = angular displacement.

4. Net Acceleration

A rotating object experiences two types of acceleration with different direction of action such that: 

1. Tangential Acceleration(a_t): tangential acceleration acts along the tangent to the object’s path, tangential acceleration and angular acceleration ($\alpha$) are related to each other such that: 

$a_t=r.\alpha$

2. Centripetal (or radial) acceleration (a_c): centripetal acceleration (a_c) is the acceleration directed towards the center of the rotation, that keeps an object in a circular path while moving.

$a_c=r.{\omega }^2$

The net acceleration is the vector sum of both tangential and centripetal accelerations. The following formula gives the total acceleration experienced by a point on a rotating object.

$a_{net}=\sqrt{a_t^2+a_c^2}$

3.0Angular Acceleration and Linear Acceleration 

Linear Acceleration 

Linear acceleration is defined as the rate of change of velocity with respect to time along a straight line. It can mathematically be represented as: 

$a=\frac{dv}{dt}$

Here, 

• a = linear acceleration 
• v =  velocity 
• t = time taken by the object. 

Relationship between Angular Acceleration and Linear Acceleration

Angular acceleration determines linear acceleration at a distance from a centre of rotation. The farther the distance from the centre, the higher the linear acceleration for the same angular acceleration. 

4.0Torque and Angular Acceleration 

Torque is the rotational equivalent of force; it is the force that causes an object to turn about an axis. The torque angular acceleration can be represented mathematically as: 

$\tau =I\alpha$

Here:

• τ = torque applied to the object.
• I = moment of inertia of the object (Moment of inertia or rotational inertia is a physical quantity that measures the resistance of an object in rotational motion about a particular axis by the formula: $I=\sum m_i{r}_i^2$)
• α = angular acceleration.

The given formula is analogous to the F = ma (force in Newton’s second law of motion for linear motion.)

Torque is responsible for angular acceleration. The larger the torque applied to an object, the greater the angular acceleration based on a constant moment of inertia. This torque-angle acceleration relationship is important in describing what happens in rotating bodies.

5.0Solved Examples 

Example 1: An object is rotating with an initial angular velocity of ω₀=2 rad/s. If the angular acceleration is α=3 rad/s², find the angular velocity after 4 seconds.

Solution: Using the first equation of motion 

$\omega ={\omega }_0+\alpha t$

$\omega =2+(3\times 4)=2+12=14rad/s$

Example 2: A wheel starts from rest and attains an angular velocity of 20 rad/s in 10 seconds. Find the angular acceleration.

Solution: by the first equation of motion 

$\omega ={\omega }_0+\alpha t$

${\omega }_0$= 0, as the wheel starts from rest, 

$20=0+\alpha \times 10$

$\alpha =\frac{20}{10}=2rad/s^2$

Example 3: A flywheel, initially at rest, is subjected to a constant torque of 500 Nm. The flywheel has a moment of inertia of 1000 kg⋅m². After 10 seconds, the angular velocity of the flywheel is 100 rad/s.

1. Find the angular acceleration of the flywheel.
2. Calculate the total work done by the torque during the 10 seconds.

Given: 

• Torque τ =500Nm
• Moment of Inertia I = 1000Kg.m²
• initial angular velocity 0=0rad/s
• Final angular velocity = 100rad/s
• Time t = 10s

Solution: 

Angular acceleration ($\alpha$)

We know $\tau =I\alpha$

$500=1000\alpha$

$\alpha =0.5ra/s^2$

Work done by torque

The work done by the torque can be determined using the work-energy theorem for rotational motion. The work done W is equal to the change in the rotational kinetic energy:

$W=\Delta K=\frac{1}{2}I({\omega }^2-{\omega }_0^2)$

$W=\frac{1}{2}I{\omega }^2$

$W=\frac{1}{2}\times 1000\times (100{)}^2$

$W=5000000J=5MJ$