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## Introduction

Hey it's a me again @drifter1! Today we continue with **Physics** and more specifically the branch "**Classical Mechanics**" to get into **Work and Power in Angular Moton**. So, without further ado, let's get straight into it!

## Work in Angular Motion

In angular motion we define the change in position as the**change of the angle**

*θ*. With that in mind think about a rigid body that

**rotates through an angle**

*dθ*from some point A to point B, while a

**force**

*F*is applied to a point P. As we described in a previous article, the position of this point is defined by the

**vector**

*r*. To make the calculation procedure simpler, let's suppose that the rigid body is

**constrained to rotate about a fixed axis**

*O*. That way vector

*r*

**moves in a circle of radius r**. The arc length

*s*of the circle if equal to

*s = θ·r*. Because vector

*r*moves in a circle, and by also defining the

**change in angle as a vector**

*θ*we can end up with the following cross product that will give us the

**change in arc length as a vector**:

From the small change in angle

*dθ*and because

*dr = 0*, from the assumption of having the rigid body fixed on O, we get:

From the definition of Work in integral form we obtain:

So, the total work done on a rigid body is equal to the

**sum of torques integrated over the angle which the body rotates**. A more descriptive way of writing the formula is:

where someone can clearly see the change from some angle A to B

If the complete body rotates through the same angle the so called

**incremental work**is equal to the sum of torque times the common incremental angle dθ:

## Work-Energy Theorem

Similarly to Translational motion, the Work-Energy Theorem also works for Angular motion. Thinking about a**rigid body rotating around a fixed axis**we have:

For

**Rolling motion**(translational + angular) the Kinetic Energy for A and B will be equal to the sum of each kinetic energy, as we described in the previous article:

## Power in Angular Motion

As you might remember, Power is equal to**rate of doing Work**. So, it's the change of work over time. Thinking about a

**constant net torque**and the simple equation of Work that we defined previously (W = τθ), the power of angular motion is given by the equation:

Examples around all the equations that we covered today and in the most articles, will be done in a separate article, when we are finished with angular motion!

## RESOURCES:

### References

- https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-_Mechanics%2C_Sound%2C_Oscillations%2C_and_Waves_(OpenStax)/10%3A_Fixed-Axis_Rotation__Introduction/10.8%3A_Work_and_Power_for_Rotational_Motion
- https://opentextbc.ca/physicstestbook2/chapter/rotational-kinetic-energy-work-and-energy-revisited/
- http://farside.ph.utexas.edu/teaching/301/lectures/node105.html

### 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, 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
- Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation

## Final words | Next up

This is actually it for today's post! Next time we will get into Angular Momentum...and the "famous" Conservation of Angular Momentum!

See ya!

Keep on drifting!