IB Sports, exercise and health science B.2.1.2 Angular Motion, Angular Momentum, and Rotational Control Notes

1. Angular motion occurs when a force is applied at a distance from the axis of rotation, causing the object to rotate instead of moving in a straight line.
2. Angular motion involves rotation around an axis, such as a gymnast spinning or a diver twisting.

Key characteristics of angular motion

1. It takes place around a fixed axis, which can be real (e.g., bicycle wheel) or imaginary (e.g., gymnast in mid-air).
2. All points on the object move in circular paths around the axis.
3. Different points on the object move at different linear speeds, but they share the same angular displacement, angular velocity, and angular acceleration.
4. The radius of the rotation affects how fast different parts move: the further from the axis, the greater the linear speed, even if the angular speed remains the same.

Angular Displacement (θ)

1. Angular displacement is a vector quantity, meaning it has both magnitude and direction.
2. It can be positive (counterclockwise) or negative (clockwise) based on the right-hand rule.
3. Angular displacement differs from angular distance. Angular displacement measures the shortest path, while angular distance includes all rotations.

Angular velocity

1. Angular velocity is a vector quantity and has both magnitude (speed of rotation) and direction.
2. The direction of angular velocity follows the right-hand rule.
3. If the rotation is counterclockwise, angular velocity is positive.
4. If it is clockwise, angular velocity is negative.

It is given by:

$$ \text{ω} = \frac{\text{θ}}{\text{t}} $$

Where:

1. ω= Angular velocity (rad/s)
2. θ= Angular displacement (rad)
3. t= Time taken (s)

Relationship between linear and angular velocity

v=r⋅ω

Where:

1. v = Linear velocity (m/s)
2. r = Radius (m)
3. ω = Angular velocity (rad/s)

Angular acceleration

It is measured in radians per second squared (rad/s²)

It is given by:

$$α= \frac{Δω}{Δt}$$

where:

1. α= Angular acceleration (rad/s²)
2. Δω= Change in angular velocity (rad/s)
3. Δt = Time interval (s)

Angular Momentum

1. Angular momentum is produced by torque when an eccentric force acts on an object free to rotate about an axis.
2. This force may come from external objects (e.g., the ground or sports equipment), another body, or muscle movements.
3. Angular momentum is a vector quantity, so it has both size and direction. It is measured in kg·m²/s.
4. It is represented by the letter L and is given by the equation: L=Iω, where I is the moment of inertia and ω is the angular velocity.
5. Angular momentum is built up when an eccentric force creates a torque, such as a gymnast pushing against the floor at an angle that does not pass through the centre of mass, so the ground reaction force creates a torque around the centre of mass and rotation builds before the athlete leaves the ground.

Conservation of Angular Momentum

1. Angular momentum remains constant unless acted upon by an external torque.
2. This principle explains why a diver can control their spin by tucking or extending their body mid-air.
3. In flight, the only forces acting on the athlete are gravity and a small amount of air resistance, and because gravity acts through the centre of mass it cannot create a torque about that axis, so airborne angular momentum is constant.
4. Because L = I × ω stays constant, if I decreases ω must increase, which is why a gymnast tucks the arms and legs in close to the centre of mass to reduce I and speed up rotation.
5. Coming out of the tuck raises I and lowers ω, slowing the rotation ready for landing, so the change in moment of inertia (not air resistance) is what changes the rate of rotation.

Transfer of Angular Momentum

1. Angular momentum cannot be created or destroyed unless an external torque is applied.
2. Once airborne, an athlete cannot change total angular momentum.
3. Internal Adjustments in Angular Velocity
   1. When one part of the body increases its angular velocity, another part must decrease its angular velocity.
   2. This maintains total angular momentum.
   3. When the first part slows down, the second part speeds up again, ensuring conservation of momentum.

How angular momentum can be redirected into twisting motion

1. Since angular momentum is a vector quantity, an athlete rotating about one axis (e.g., during a somersault) can introduce rotation around another axis (e.g., tilting) by adjusting body segment movements using muscles.
2. The combination of these two vectors results in rotation around a third axis, creating twisting motion. In gymnastics, this principle is used to generate twists during somersaults.
3. Because angular momentum is a vector, it can also be traded between rotation axes, so a gymnast somersaulting around one axis can introduce a small tilt with the trunk and arms and convert some of the somersault angular momentum into twisting around a different axis, which is how high-difficulty diving and gymnastic skills combine somersaults with twists in mid-air.

Torque Produces Angular Motion

1. A torque (also called moment of force) is what a force does to a body that is free to rotate, as long as the line of the force does not pass through the axis of rotation.
2. A force whose line of action misses the axis is called an eccentric force, and it is the only kind of force that creates torque.
3. The size of the torque depends on the size of the force, the direction in which it is applied, and the perpendicular distance from the axis of rotation (called the moment arm).
4. The formula is τ = F × d, measured in newton metres (N m).

Moment of inertia affects how easily a body rotates

1. It is expressed in kg·m². It depends on two key factors: mass of the object or body and distribution of mass relative to the axis of rotation.
2. Greater mass farther from the axis increases the moment of inertia, making rotation more difficult.
3. Mass closer to the axis decreases the moment of inertia, making rotation easier.
4. The human body’s moment of inertia varies depending on body position and axis of rotation.