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 accelerationIn 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
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 ti = 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 ti to zero again and rearranging we obtain:
This equation looks very similar to the linear kinematic equation vf = vi + 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 ti to zero and afterwards rearranging we get:
An equation very similar to the equation for linear motion: sf = si + uit + 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: vf2 = vi2 + 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_2Consider 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.
- 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 (ωi = 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 ωi = 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!
Mathematical equations used in this article, where made using quicklatex.
Previous articles of the series
- 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
- 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 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...
Keep on drifting!