(1) θ = θ_i + ω_i *t + (1/2)α * t^2
(2) ω = ω_i + α * t
where * means times,
^2 means "squared",
θ_i is the initial angular position, θ is the angular position at time t,
ω_i is the initial angular velocity, ω is the angular velocity at time t, and α
is the (constant) angular acceleration.
From (1) and (2), it can be shown that:
(3) θ - θ_i = (1/2)(ω_i + ω)
(4) ω^2 = ω_i ^2 + 2α(θ - θ_i)
-This page is "kinematics" since it does not talk why the objects are rotating. If we included torques and moments of inertia we would be talking about "dynamics".
-There is a complete analogy between the formulas for constant acceleration in one dimension and the formulas for constant angular acceleration around one axis.
-The table below shows which variables for motion along one dimension correspond to which variables for rotation around one axis. Time t plays the same role for both types of motion, so it is listed in both columns.
If you start with a formula for constant acceleration in one dimension you can get a formula for constant angular acceleration around one axis by replacing each variable in it with the corresponding variable for rotation around one axis, using the table above. Some examples of this are listed in the table below.
Some corresponding equations
Linear Motion
x = x_i + v_i * t + (1/2)at2
v = v_i + at
Angular motion
θ = θ_i + ω_i *t + (1/2)α * t^2
ω = ω_i + α * t
-This correspondence can also be extended to one dimensional "dynamics". However, it doesn't work in three dimensions; rotation in three dimensions is more complicated than linear motion in three dimensions.
A fan, starting from rest, begins to turn. Its angular acceleration is 0.500 radian/ second for 5.00 seconds. At the end of those 5 seconds, what is its angular velocity, and what total angle has it turned through?
Solution Since the fan is starting from rest, we its initial angular velocity ω_i is zero. The time t is 5 seconds. The angular acceleration α is 0.5 rad / s^2.
Now equation (2) for angular velocity ω
becomes a plug-in:
ω = ω_i + α * t = 0 + (0.5 rad/s^2)(5 s) = 2.5 rad/s.
For convenience, we can choose the zero of angle to be the starting position, so that θ_i, the initial angular position, is zero. Now equation (1) is also a plug-in.
So θ = θ_i + ω_i *t + (1/2)α * t^2 = 0 + 0 * (5 s) + (1/2)(0.5 rad/s^2)( 5 s)^2 = 6.25 radians.
Many people are more comfortable with degrees than with radians.
1 radian = 180/π degrees.
Dividing both sides of this equation by (1 radian) gives
1 = (180/π) (degrees/radian).
You can multiply anything by 1 without changing it, so
θ = 6.25 rad = (6.25 rad) 1 = (6.25 rad)(180/π) (degrees/radian) ~ 358 degrees.
Similarly, ω = 2.5 rad/s = (2.5 rad/s)()(180/π) (degrees/radian) ~ 143 degrees/s,
and α = 0.5 rad/s^2 = (0.5 rad/s^2) )(180/π) (degrees/radian) ~ 28.6 degrees/s^2.
-The method we just used to convert units is called the "factor-label method".
-If we had done the calculation in degrees we would have gotten the same answer.