Angular Kinematics

Angular motion

motion around an axis of rotation; ex: spins, swings, or twists

Radians

ratio between the length of an arc and its radius in a given circle, rad = arc length/ radius.

Degrees to radian

degrees/57.3= rad

radians to degrees

57.3 x rad = degrees

Absolute angles

angle between a segment and the right horizontal of the distal end, measure angle from horizontal line in a ccw direction

Relative angles

joint angles, angle between the longitudinal axis of two adjacent segments; fully extended position is defined as either 0 or 180 degrees.

Relative angle of a hip joint

flexion: +

extension:-

relative angle of a knee joint

flexion:+

extension:-

relative angle of an ankle joint

dorsiflexion:+

plantarflexion:-

Calculating absolute angles

θ = 𝐭𝐚𝐧−𝟏(y proximal – y distal/ x proximal – x distal )

Absolute angle quadrant 1

+x, +y

Absolute angle quadrant 2

-x, +y, add 180 degrees to absolute angle

Absolute angle quadrant 3

-x, -y, add 180 to absolute angle

Absolute angle quadrant 4

+x, -y, add 360 to absolute angle

Law of Cosines

used if you have segment lengths, or points for three joints; 1st calculate the length of each side then calculate the cosine theta angle.

Two absolute angles

find relative angles using absolute angles, θrelative= 𝛉 absolute leg + (180 - 𝛉absolute thigh)

Relative angle hip

𝛉Hip = θ trunk +(180 −θthigh)

Relative angle knee

𝛉𝐊𝐧𝐞𝐞 =θshank + (180 −θthigh)

Relative angle of ankle

𝛉Ankle = θ shank + (180 – θ foot)

Right-thumb rule

• curled fingers of the right-hand point in the direction of the rotation

• direction of the extended thumb point to the direction of the vector

• +=ccw

• -=cw

Angular displacement

change in angular position, ∆𝜃 = 𝜃𝑓 − 𝜃i, 𝜃 must be in radian

If there us more than one rotation, ∆𝜃= 𝜃𝑓−𝜃𝑖 +𝑛×360°

Angular distance

the sum of the magnitude of all angular changes undergone by rotating body

Angular velocity (ω)

how fast the angle is changing, the vector change of the angular displacement.

𝜔 = ∆𝜃/∆t

Tangential velocity

the instantaneous linear velocity of an object in rotational motion.

𝑣𝑡 = 𝑟ω

Angular acceleration

rate of change of angular velocity, 𝛼 = ∆ω /∆t

Tangential acceleration

the linear acceleration resulting from angular acceleration changing linear velocity and is only valid if angular acceleration is expressed in rad/s²,

a𝑡 = 𝑟α

Centripetal radial acceleration

inward acceleration necessary for angular motion, it does not change the speed of an object just maintains the objects circular motion.

ac =𝑣𝑡²/r or ac=ω²r