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A particle moves in a circle of radius r. Having moved an arc
length s, its angular position is θ relative to its original position,
where .
An angular displacement is a change in
angular position:
where Δθ is the angular displacement, θ1 is the initial angular position and θ2 is the
final angular position.
Using the formula for angular position and letting
,
we have also
-Angular velocity is the change in angular displacement per unit time. The symbol
for angular velocity is ω and the units are typically rad s-1. Angular speed is the
magnitude of angular velocity.
-The instantaneous
angular velocity is
given by
-Using the formula for angular position and letting
we have also
where v is the translational speed of the particle.
-Angular velocity and frequency are related by
A changing angular velocity indicates the presence of an angular acceleration in rigid body, typically measured in rad s −2. The average angular acceleration
over a time interval Δt
A changing angular velocity indicates the presence of an angular acceleration in
rigid body, typically measured in rad s−2. The average angular acceleration over a
time interval Δt is given by
.
The instantaneous acceleration α(t) is given by
Thus, the angular acceleration is the rate of change of the
angular velocity, just as acceleration is the rate of change of velocity.
The translational acceleration of a point on the object rotating
is given by
The radial acceleration (perpendicular to direction
of motion) is given by
I = mr2.
Torque τ is the twisting effect of a force F applied to a rotating object which is at
position r from its axis of rotation. Mathematically,
where r is the particle's position from the origin, p = mv is
its linear momentum, and × denotes the cross product.