Introduction to Angular Motion
In our previous chapters, we looked at how athletes move in straight lines. But in sport, movement is rarely just linear. Think of a gymnast performing a backflip, a diver spinning through the air, or a discus thrower rotating in the circle. This is angular motion—movement around a fixed point or axis.
Understanding the physics behind these spins can be the difference between a gold medal and a fall. Don't worry if the math sounds intimidating; we are going to break it down into simple, logical steps that apply to every athlete you see on TV.
Note: This chapter builds on what you know from "Biomechanical principles" regarding Newton's Laws and "Linear motion."
1. Describing How We Spin
Just like linear motion has distance and speed, angular motion has its own set of measurements. Instead of meters and kilometers, we use radians (\(rad\)) to measure angles.
Angular Displacement
Angular displacement is the smallest change in angle between the starting and finishing point of a rotating body. Imagine a clock hand moving from 12 to 3; that change in position is its displacement.
Unit: Radians (\(rad\))
Angular Velocity
This is simply how fast something is spinning. It is the rate of change of angular displacement.
Formula: \(Angular\ Velocity = \frac{Angular\ Displacement}{Time}\)
Unit: Radians per second (\(rad/s\))
Angular Acceleration
This measures how much the spinning speed is changing. If a figure skater pulls their arms in and starts spinning faster, they are experiencing angular acceleration.
Formula: \(Angular\ Acceleration = \frac{Change\ in\ Angular\ Velocity}{Time}\)
Unit: Radians per second squared (\(rad/s^2\))
Quick Review: Think of a wheel. Displacement is how far it turned, velocity is how fast it’s turning right now, and acceleration is whether it's speeding up or slowing down its spin.
2. Newton’s Laws Applied to Angular Motion
Isaac Newton’s laws don't just apply to things moving in straight lines; they apply to rotations too! To understand these, we need to know about Torque (also known as eccentric force)—this is the "turning force" that makes things rotate.
Newton’s First Law (The Law of Inertia)
A rotating body will continue to turn about its axis with constant angular momentum unless acted upon by an external torque.
In simple terms: An ice skater will keep spinning at the same rate until friction from the ice or a change in body position (internal force creating torque) stops them.
Newton’s Second Law (The Law of Acceleration)
The angular acceleration of a body is proportional to the torque applied and takes place in the direction in which the torque acts.
In simple terms: The harder you push (torque) on a swing, the faster it will accelerate into its arc.
Newton’s Third Law (The Law of Action and Reaction)
For every torque applied by one body on another, there is an equal and opposite torque applied by the latter on the former.
In simple terms: When a diver jumps off a board, they push down on the board with a turning force, and the board pushes back with the same force, helping them flip.
3. Moment of Inertia (MI)
Moment of Inertia is a fancy way of saying "how hard it is to get something spinning" or "resistance to rotation."
It depends on two main things:
1. Mass: The heavier the object, the harder it is to spin.
2. Distribution of mass from the axis: This is the most important part for athletes! The further the mass is from the point of rotation, the higher the Moment of Inertia, and the harder it is to spin.
The "Tuck" Example:
Imagine a diver.
- When they are in a straight position, their mass is far away from their center. This means they have a High Moment of Inertia (they resist spinning).
- When they pull into a tight tuck, they bring their mass close to the axis. This creates a Low Moment of Inertia (it is very easy to spin).
4. Angular Momentum
Angular Momentum is the total "quantity of rotation" a body has.
Formula: \(Angular\ Momentum = Moment\ of\ Inertia \times Angular\ Velocity\)
Conservation of Angular Momentum
This is a vital rule for your exams: Angular momentum remains constant (conserved) while a body is in flight.
Once an athlete leaves the ground (like a high jumper or a gymnast), they cannot change their total angular momentum. However, they can change the two parts of the equation (Moment of Inertia and Angular Velocity) to control their spin.
Did you know? Because the total stays the same, if Moment of Inertia goes down, Angular Velocity must go up to compensate!
5. The Relationship Between MI and Velocity
Since \(Angular\ Momentum\) is fixed during flight, athletes use a "trade-off" system to perform skills.
To Spin Faster (e.g., during the middle of a somersault):
1. The athlete moves their mass closer to the axis (tucks their knees).
2. This decreases Moment of Inertia.
3. To keep angular momentum constant, Angular Velocity increases.
4. Result: The athlete spins rapidly.
To Spin Slower (e.g., when preparing to land):
1. The athlete opens up their body (extends arms and legs).
2. This increases Moment of Inertia (mass is further from the axis).
3. To keep angular momentum constant, Angular Velocity decreases.
4. Result: The spin slows down, allowing for a safe, controlled landing.
Common Mistake to Avoid: Many students think the athlete "gains" momentum when they tuck. They don't! The momentum stays the same; it is the speed (velocity) that changes because the resistance (MI) decreased.
Key Takeaways for Revision
1. Angular Displacement: Change in angle (\(rad\)).
2. Angular Velocity: Speed of spin (\(rad/s\)).
3. Moment of Inertia (MI): Resistance to spin. Higher when mass is spread out.
4. Angular Momentum: Total spin quantity (\(MI \times Velocity\)). It stays the same during flight!
5. The Trade-off: If you want to spin faster, decrease your MI (tuck). If you want to slow down, increase your MI (spread out).