Rotational Kinematics Torque, and rolling motion

Related formulas

The kinematics of angular motion is set just like the kinematics of linear motion. We have the following relationships:
For angular speed: ω = ω_o + α t
ω is the angular velocity = dθ/dt, ω_o is the initial angular velocity (at time equal zero), and α is angular acceleration = dω/dt = d²θ/dt²

For angular position:

θ(t) = θ_o + ω_ot + (1/2) αt²
θ_o is the initial angular position of the rotating point.

We have also the following relationship:
ω² - ω_o² = 2α(θ - θ_o)

The way of decribing angular distance and speed in degrees is not convenient, but it is in radian. It is easier to write an angle is radian than in degrees (100π = 36000^o) For small angles, if θ is measured in radians, we can approximate sin θ or tg θ by θ . The intersting related formula is that the circumference of a circle is 2π times the radius. Dividing a circle in 2π parts gives a part called radian. A radian is equal to the an arc that has the measure of the related radius.

An object that moves from an angle θ_i at t_i to an angle θ_f has average angular speed
ω = (θ _f - θ _i)/(t_f - t_i) = Δθ/Δt.

Instantaneous angular speed is obtained by taking Δt small, and write:
ω = dθ/dt = lim Δθ/Δt
when Δt tends to dt.

If an object moves (rotates) 100 rpm (revolutions per minute), it will have 2π x 100 radians/minute = 10.50 rad/s. ω is measured in rad/seconds.

Recall an object moving in a circle (circular motion) at constant speed (uniform), has no tangential acceleration. With no constant speed, we can write:
v = ω r, hence
a = dv/dt = rdω/dt + ω dr/dt = α r + ω v = α r + ω² r

The first term is the tangential acceleration and
the second the radial or centripetal acceleration.

We have the following vectorial relations:

a = a_t + a_c
a_t = α r
a_c = ω² r

To recap: