ROTATIONAL
KINEMATICS
A. Rotation of a Rigid body about a fixed
axis
1)Rotational kinematic parameters (q,w,a)
and their relationships with linear kinematic parameters (x,v,a).
Table 1: Linear quantities and their
rotational counterparts
|
Linear |
Rotational |
Relationship |
| displacement |
Dx (m; vector) |
Dq (rad;scalar) |
x = rq |
| velocity |
v (m/s; vector) |
w (rad/s;vector) |
v = rw |
| acceleration |
a (m/s2; vector) |
a (rad/s2;
vector) |
a = ra |
Table 2: Motion with constant acceleration
| velocity, time |
v = vo + at |
w = wo
+ at |
| displacement. time |
x = xo + vot + (1/2)at2 |
q = qo
+ wot
+ (1/2)at2 |
| velocity, displacement |
v2 = vo2
+ 2a(x - xo) |
w2
= wo2
+ 2a(q
- qo) |
2) Rotational Kinetic Energy and its relationship
with linear kinetic energy + the definition on moment of inertia
-
Linear Kinetic Energy: K = (1/2)mv2
-
Rotational Kinetic Energy: K = (1/2)Iw2
-
Moment of Inertia: the measure of an object's
resistance to rotational motion. It is a scalar quantity with units
of kg m2. The value for the moment of inertia is dependent
on the choice of axis of rotation, therefore its value may differ depending
on the choice of this axis.
-
For a system of masses: I = Smiri2
where ri is the distance between the mass and the axis of interest.
-
For a continuous object or a rigid body:
-
Below is a list of moment of inertia of a few
common objects:
-
Paralle-Axis Theorem: If we know the moment of
inertia about the CM, the parallel axis theorem helps us find the moment
of inertia about any other axis: I' = Icm + MD2
where D is the distance between the CM axis and the new axis.