Angular Motion

Angular Motion and Eccentric Force

Angular motion is movement/rotation around a fixed point or axis, and appears constantly in sport – either as whole-body rotation (e.g. a somersault) or as rotation of individual body parts around a joint (e.g. the arm rotating at the shoulder when bowling in cricket).

• Angular motion is produced by an eccentric force – a force applied away from an object's centre of mass, which causes it to turn rather than travel in a straight line. A force applied directly through the centre of mass (a concentric force) instead produces pure linear motion with no rotation
• Even sports that look purely linear, such as sprinting, contain a large angular component: the arms and legs continuously rotate around the shoulder and hip joints as the athlete runs forward in a straight line

Axes of Rotation

The body can rotate around one of three axes, depending on the direction of the eccentric force applied:

• The transverse axis runs side to side through the body – rotation about this axis produces movements such as a forward or backward somersault
• The sagittal axis runs front to back through the body – rotation about this axis produces movements such as a cartwheel
• The longitudinal axis runs top to bottom through the body – rotation about this axis produces movements such as a spin in figure skating

Torque

Torque (also called a moment) is the rotational effect of a force – the turning effect that causes a body to rotate about its axis.

• Torque is increased either by applying a larger force, or by applying the same force further from the axis of rotation (a longer moment arm) – e.g. pushing a door open near its hinge requires much more force than pushing the same door open at its outer edge, because the perpendicular distance from the hinge (the pivot) is far greater at the edge
• Torque is calculated as:

Newton's Laws Applied to Angular Motion

Newton's three laws of motion can be reapplied to rotational movement by substituting force for torque and momentum for angular momentum.

• Newton's first law (angular) – a rotating body continues to spin about its axis with constant angular momentum unless an external torque acts upon it – e.g. an ice skater performing a jump keeps spinning at a constant rate throughout the flight phase, and it is only the torque generated when their skates strike the ice that changes their rotational motion
• Newton's second law (angular) – the rate of change of a body's angular momentum is proportional to the torque causing it, and takes place in the direction that torque acts – in practice, a larger torque produces a faster rotation
• Newton's third law (angular) – for every torque applied by one body to another, an equal and opposite torque is exerted back – e.g. when a goalkeeper throws both arms upward to tip a shot over the bar, this generates an equal and opposite reaction that swings their lower legs backward

Angular Displacement, Velocity and Acceleration

• Angular displacement is the smallest angle of turn between a body's starting and finishing position during a rotation, measured in degrees or in radians (1 radian = 57.3 degrees)
• Angular velocity is a vector quantity describing rotational speed together with the axis being rotated around:

• Angular acceleration is the rate of change of angular velocity:

Moment of Inertia

Moment of inertia is a body's resistance to angular motion – resistance both to starting a rotation and, once rotating, to having that rotation stopped (much like a revolving door: hard to get moving, then equally hard to bring to a stop once spinning). It depends on two factors:

• Mass – a greater mass increases moment of inertia and therefore resistance to rotation, e.g. a bowling ball is harder to start rolling than a football, but is equally harder to stop once moving
• Distribution of mass relative to the axis of rotation – the further a body's mass is spread from its axis of rotation, the greater its moment of inertia. This is why an open (straight-body) somersault, where the diver's mass is extended away from their axis, has a higher moment of inertia and is harder to perform than a tucked somersault, where the mass is pulled in close to the axis
• This principle applies within a single stride of sprinting too: the recovery leg is drawn in with a high knee lift, bringing its mass closer to the hip joint and lowering its moment of inertia so it can swing through faster, ready for the next powerful drive phase

Angular Momentum and Conservation

Angular momentum is the quantity of rotation, or 'spin', a body possesses:

Moment of inertia and angular velocity are inversely proportional – if one increases, the other must decrease to keep angular momentum unchanged.

• Angular momentum is conserved (stays constant) whenever no external torque acts on a rotating body, which applies during flight or on very low-friction surfaces like ice – e.g. once a diver leaves the board, their overall angular momentum remains fixed throughout the dive and can only be changed again by an external torque, such as hitting the water
• Because angular momentum is fixed during a movement, a performer can only change their rotational speed by deliberately altering their moment of inertia – e.g. a figure skater starts a spin with arms and one leg extended outward (large moment of inertia, slower rotation), then pulls the limbs in tight to the body (small moment of inertia), which sharply increases angular velocity and produces a much faster spin
• The same principle explains why a diver often extends their body out of a tuck just before entering the water: increasing their moment of inertia at that point slows their rotation down, helping them straighten out and control their entry