Physics - Classical Mechanics - Rotation about a moving axis (Rolli...

Introduction

Hey it's a me again @drifter1! Today we continue with Physics and more specifically the branch "Classical Mechanics" to get into the Rotation about a moving axis, which can also be called Rolling motion. So, without further ado, let's get straight into it!

Fixed Axis Rotation

Until now we mostly talked about Translational and Angular motion separately. So, an object either moved from one point to another in space (translational motion) or rotated about it's axis that was also a fixed point (angular motion). The first part has been covered a lot in earlier articles of the series. The second part is what we are covering right now in the series. Combining the terms and quantities that we studied in the previous articles about Angular motion, right now we know that an object rotates:

around a fixed axis of rotation 'O', called the pivot point
because of a change in it's angular velocity (ω) that has been caused by a Torque (τ) that made it accelerate with some angular acceleration (a). Also think about: Στ = Ia
wanting to keep it's "rotation state" (rotating or not)

If an object simply rotates around a fixed axis then we can define it's total kinetic energy to be equal to it's angular kinetic energy. Let's consider a rigid body with some mass m and radius R and that we want to calculate the kinetic energy at the point of contact to the ground (will help us transition to rolling motion). Of course, we have to apply the Parallel Axis Theorem to get the Moment of Inertia about the point of contact. So, the angular kinetic energy is given by:

Moving Axis Rotation

Let's now think about the combined motion of translation + rotation (rolling motion). Continuing with the expansion of the previous equation we get:

So, you can see the that kinetic energy at the point of contact is equal to the sum of Angular/Rotational and Translational kinetic energy. In this previous equation the translational motion part is of course equal to the kinetic energy of the center-of-mass (CM) and so we can also write that the total kinetic energy of a rolling body is:

This means that calculating the kinetic energy of a rolling body is quite simple! We just have to calculate the kinetic energy of it's center-of-mass and add it to the angular kinetic energy in respect to the center-of-mass. This means that we only have to know the moment of Inertia (I) about the center-of-mass, which can be found easily for common shapes.

Of course, the previous equation can be quite misleading and that's why I made the complete proof using the moment of inertia around the point of contact, where we also had to apply the Parallel Axis Theorem. We never really added the impact of the translational motion, we just tried to calculate the kinetic energy at some point different than the center-of-mass (and mostly axis of rotation). Talking about a distance of R the substitution of the term ωR gave us the tangential velocity v (which is also equal to the velocity of the center-of-mass) and so, in the end, the kinetic energy of the center-of-mass (K_cm = ½mv²). Stay Cautious!

RESOURCES:

References

Images

https://knowyourmeme.com/photos/201808-they-see-me-rollin

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, acceleration and averages of those
Rectlinear motion with constant acceleration and free falling -> const acceleration 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 acceleration 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
Internal Energy and Work -> 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
Rotation with constant angular acceleration -> Constant angular acceleration, Example
Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
Parallel Axis Theorem -> Parallel axis theorem with example
Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example

Final words | Next up

This is actually it for today's post! Next time we will continue with the definition of Work and Power in Angular Motion....

See ya!

Keep on drifting!

Physics - Classical Mechanics - Rotation about a moving axis (Rolling motion)