Physics - Classical Mechanics - Rotation with constant angular acce...

Introduction

Hey it's a me again @drifter1! Today we continue with Physics and more specifically the branch "Classical Mechanics" to get into Rotations with constant acceleration. So, without further ado, let's get straight into it!

Constant angular acceleration

In the previous article we described the basic rotational quantities θ, ω and a and how they are related to each other. For particles in rotational (or angular) motion we define:

θ as the position or angular angle
ω as the angular velocity
a as the angular acceleration

These quantities are related to each other as following:

The average angular velocity is simply equal to half the sum of the initial and final values:

Changing the d's with Δ's (and so talking about change of angle over change of time), in the first equation for ω, we can calculate the average angular velocity also with:

Solving for θ and setting t_i = 0 we get:

Thinking about the constant angular acceleration over some period of time (initial to final) and angular velocity from some initial to some final value, we can write the following integral:

Setting the initial time t_i to zero again and rearranging we obtain:

This equation looks very similar to the linear kinematic equation v_f = v_i + at.
I guess you can already foresee that the equation for the position/angle will also look quite similar! Doing similar stuff with the equation for angular velocity we can easily find an equation that gives us the final angle after some period of time, when knowing the initial angle and velocity and also the constant angular acceleration of the motion. So, starting off with the equation for angular velocity and substituting what we obtained before we get:

Again setting t_i to zero and afterwards rearranging we get:

An equation very similar to the equation for linear motion: s_f = s_i + u_it + 1/2at².
Solving the equation for angular velocity for time t and substituting it into the equation for angular angle, we can get an equation that's independent of time (I will spare the calculations this time):

Yet another equation similar to liner motion's: v_f² = v_i² + 2a (Δs).

So, by doing all this we ended up with 4 very useful equation that let us calculate:

Angular displacement from average angular velocity
Angular velocity from angular acceleration
Angular displacement from angular velocity and angular acceleration
Angular velocity from angular displacement and angular acceleration

Example from Reference_2

Consider a fishing reel that's initially at rest. The fishing line unwinds from the reel at a radius of 4.5cm from it's axis of rotation (suppose that this radius doesn't change as it unwinds). The reel is given a constant angular acceleration of 110 rad/s² for 2s.
Calculate:

The final angular velocity of the reel after 2s
How many revolutions the reel makes

1.

We know the acceleration a and time t and want to determine ω. For this we can use the "second" equation. Since the initial angular velocity is zero (ω_i = 0) the final angular velocity is the only unknown quantity. Therefore we have:

2.

A revolutions is 1 rev = 2π rad and so to find the number of revolutions we first have to find the θ in radians. We know the values of a and t and that ω_i = 0, so we can obtain θ easily using the "third" equation:

The number of revolutions is:

As you can see, examples around constant angular acceleration are quite simple, as we just have to apply the 4 kinematic equation that we proved today!

RESOURCES:

References

Images

Mathematical equations used in this article, where made using quicklatex.

Previous articles of the series

Rectlinear motion

Velocity and acceleration in a rectlinear motion -> velocity, accelaration and averages of those
Rectlinear motion with constant accelaration and free falling -> const accelaration motion and free fall
Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
Rectlinear motion exercises -> examples and tasks in rectlinear motion

Plane motion

Position, velocity and acceleration vectors in a plane motion -> position, velocity and accelaration in plane motion
Projectile motion as a plane motion -> missile/bullet motion as a plane motion
Smooth Circular motion -> smooth circular motion theory
Plane motion exercises -> examples and tasks in plane motions

Newton's laws and Applications

Force and Newton's first law -> force, 1st law
Mass and Newton's second law -> mass, 2nd law
Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
Contact forces and friction -> contact force, friction
Dynamics of Circular motion -> circular motion dynamics, applications
Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
Contact force and friction examples -> exercises in force and friction
Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
Advanced Newton law examples -> advanced (more difficult) exercises

Work and Energy

Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
Potential Energy Diagrams -> Internal Energy, Internal Work

Momentum and Impulse

Conservation of Momentum -> Momentum, Conservation of Momentum
Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
Collision Examples -> Various Elastic and Inelastic Collision Examples
Impulse -> Impulse with Example
Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis

Angular Motion

Angular motion basics -> Angular position, velocity and acceleration

Final words | Next up

This is actually it for today's post! Next time we will get into Angular Kinetic Energy...

See ya!

Keep on drifting!

Physics - Classical Mechanics - Rotation with constant angular acceleration