Angular Kinematics

Chapter 6

KINE 3301Biomechanics of Human Movement

Radian

A radian is a ratio variable. The arc length (s) is divided by the radius (r).

Segment Angles & Joint Angles

A segment angle is the angle from the right horizontal to the segment.

A joint angle is the angle between two segments.

Angular Variables & Right Hand Rule

Right Hand Rule: Curl the fingers of your right hand in the direction of rotation and your thumb points in the direction of the angular motion vector.

Angular Velocity

𝜔=𝜃 𝑓 −𝜃 𝑖

𝑡

Angular velocity is the rate of change of the angular position, or the slope of the angle – time curve. The units for angular velocity are r/s.

The direction of the angular velocity vector is defined by the right hand rule.

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Angular Acceleration

𝛼=𝜔 𝑓 −𝜔𝑖

𝑡Angular acceleration is the rate of change in angular velocity. The units for angular acceleration are r/s2.

The direction of the angular acceleration vector is defined by the right hand rule.

𝑣=𝜔𝑟

The relationship between linear and angular velocity is defined by the equation below. The angular velocity must be in r/s.

Tangential & Radial Acceleration

An object rotating has two linear accelerations. Tangential acceleration is tangent to the path and centripetal acceleration is directed toward the center of rotation. The units for ac and aT are m/s2.

Tangential Acceleration

The tangential acceleration represents the acceleration necessary to change the rate of rotation. It is tangent to the path with units of m/s2. If the object is rotating at a constant velocity tangential acceleration is zero.

Centripetal Acceleration

Centripetal acceleration is directed inward towards the center of the circle. To keep an object rotating in a circle it must be accelerated with a centripetal acceleration. The units for centripetal acceleration are m/s2.

Centripetal Force

If you multiply the centripetal acceleration by mass you get the centripetal force. The centripetal force is the force necessary to keep an object rotating in a circle, it has units of N and like centripetal acceleration it is directed inward toward the center of the circle.

A volleyball player’s forearm rotates from an initial angle ) of .4 r to a final angle ) of .9 r in t = .2 seconds, compute the angular velocity ().

If the radius r of rotation for the volleyball players arm about the shoulder is .72 m what is the linear velocity of the hand?

A baseball player’s forearm rotates from an initial angular velocity ) of 1.4 r/s to a final angular velocity ) of .9 r/s in t = .2 seconds, compute the angular acceleration ().

If the radius r of rotation for the baseball players arm about the shoulder is .83 m what is the tangential acceleration (aT)?

Compute the centripetal acceleration (aC) and centripetal force (Fc) necessary to keep a 2 kg discus rotating with a radius of 1.3 m and an angular velocity () of 2.2 r/s.

Compute the centripetal acceleration (aC) and centripetal force (Fc) necessary to keep a 7.257 kg hammer rotating with a radius of 2.1 m and a linear velocity (v) of 13 m/s.

What is the angular velocity of the hammer?

A soccer player accelerates the lower leg with an angular acceleration () of −14 r/s2 for t = 0.3 seconds, if the initial angular velocity ) was 0.3 r/s, what was the final angular velocity ) ?

Using the final angular velocity ) above and a radius of 1.34 m what was the linear velocity of the foot?

A track athlete increases her velocity from Vi = 7.8 m/s to Vf = 8.4 m/s in a time of 0.8 s, what was the tangential acceleration (aT)?

Compute centripetal force (Fc) necessary to swing a 7.6 kg bowling ball with a velocity of 12 m/s and a radius of 1.2 m.