System of Particles and Rotational Motion · Lesson 6 of 15
Angular Velocity, Angular Acceleration and Linear Motion
“Connect the common angular motion of a rigid body with the velocities of its different particles.”
• Define angular velocity and acceleration. • Use v = ωr with the perpendicular axis distance. • Derive the vector relation v = ω × r. • Distinguish tangential acceleration from radial acceleration.
One angular motion, different linear speeds
For rotation about a fixed axis, choose an angular coordinate θ measured in radians from a reference direction. Over a finite time, average angular velocity is Δθ/Δt. Its instantaneous value is dθ/dt. Every point of a rigid body sweeps through the same angular displacement during a given interval. A point farther from the axis follows a larger circular arc and therefore has a greater linear speed, even though the angular speed is shared.
The speed relation follows by differentiating arc length s = r⊥θ for a fixed particle radius. Treat angular velocity as a vector along the axis using the right-hand rule. Anticlockwise rotation as viewed from the positive end of the axis gives a vector toward that end. Reversing the spin reverses the vector. For a fixed axis, the signed scalar ω describes the same information more simply.
Deriving the vector relation
Take an origin O on the rotation axis. Split the particle position OP into OC, along the axis, and CP, perpendicular from the axis to the particle. The cross product ω × OC vanishes because those vectors are parallel. The remaining product ω × CP is tangential, has magnitude ωr⊥ and points along the particle’s motion. It is therefore precisely the linear velocity.
Problem
A wheel rotates at 10 rad/s. Points lie 0.10 m and 0.30 m from its axle.
- 1.Their speeds are 10(0.10) = 1 m/s and 10(0.30) = 3 m/s.
- 2.Their angular velocities are identical, while their tangential velocities differ in magnitude and, at different angular positions, in direction.
- 3.A point on the axle has zero linear speed in the fixed-axis frame.
Angular acceleration and changing direction
Angular acceleration is dω/dt. For a fixed axis, its direction is along or opposite that axis according to the sign of angular-speed change. If the axis moves, angular acceleration can also reflect a changing direction of ω. A constant angular speed about a fixed axis does not imply zero linear acceleration of a rim point: its velocity direction keeps changing.
The radial component and tangential component are perpendicular. This circular-motion connection prevents confusing angular acceleration with the whole linear acceleration. For rotation combined with translation, a point velocity can be written as CM velocity plus the rotational contribution about the CM. In general, finite rotations about different axes cannot be manipulated as ordinary additive displacement vectors, even though instantaneous angular velocity is a vector.
Quiz
Which quantity is shared by all points of a rigid wheel?
All points undergo the same angular motion about the fixed axis.
Which radius belongs in v = ωr?
Each point’s circular path is centred on the axis.
Constant nonzero ω gives a rim point which acceleration?
Velocity direction changes even at constant speed.
The vector ω × r points where?
It is perpendicular to both axis direction and position vector.
For a fixed axis, α measures what?
It is angular acceleration, measured in rad/s².
Key Takeaways
• All body points share angular velocity, not linear velocity. • Use perpendicular axis distance in speed calculations. • The cross product supplies both direction and magnitude of linear velocity. • Constant angular speed still gives radial linear acceleration.