Rotational Kinematics
Rotational kinematics investigates lows of motion of objects along circular path without any reference to forces that cause the motion to change
Here
is initial time on time interval
in seconds
is final time
is original angular position at time
in radians
is final angular position at time
is angular displacement during time
is original linear velocity vector at time
tangent to trajectory
is final linear velocity vector at time
, tangent to trajectory
is total linear acceleration vector, in plane of motion
is normal component of total acceleration vector
is tangential component of total acceleration vector
is radius vector of trajectory, normal to trajectory
is length of path during time
is angular velocity vector, normal to plane of motion
is angular acceleration vector, normal to plane of motion
General formulas
Angular displacement
Average angular speed on time interval
Instantaneous angular speed at time
Original angular speed at time
Instantaneous angular velocity vector
where
is unit vector of z-axis shown in the above diagram
Instantaneous angular acceleration vector
Linear velocity vector is defined by vector product, where
is perpendicual to
Magnitude of linear velocity
Magnitude of normal component of total linear acceleration
Magnitude of tangential component of total linear acceleration
Total linear acceleration vector is defined by vector sum
Magnitude of total linear acceleration
Angle between linear acceleration and linear velocity vectors
Relation between angular acceleration and angular velocity
Kinematic equation for rotation
Uniform rotation
Uniformly accelerated rotation
At
the rotation is accelerated
At
the rotation is decelerated
Rotational Dynamics
Rotational dynamics investigates rotational motion of objects and deals with effects that forces have on motion
Rotation of point particle
Here
m is mass of the particle moving in x-y plane
is force vector applied in the plane of motion
is velocity vector, tangent to trajectory
is linear momentum vector, parallel to
is radius vector of curvature of trajectory, normal to trajectory
is angular velocity vector, normal to plane of motion
is angular acceleration vector, normal to plane of motion
is angular momentum, parallel to
is torque associated with the force
, normal to plane of motion
d is level arm of
General formulas
Moment of inertia of the particle about center of rotation
Angular momentum vector is defined by vector product
where
is linear momentum vector, perpendicular to
Relation between angular momentum and angular velocity vectors
The magnitude of angular momentum
Torque vector is defined by vector product
The magnitude of torque
where:
is angle between vectors
and
, shown in the above diagram
is level arm (or moment arm) of
Newton's Second Law in angular form:
- for general case
- for constant moment of inertia
Plane rotation of symmetric solid about its axis of symmetry
Moment of inertia of the solid about axis of rotation
where
mi is small portion of mass number i at distance Ri between its center and axis of rotation (for i = 1, 2, 3, ... , n)
dV is infinitesimal volume with density
at distance
R from axis of rotation
Parallel Axis Theorem
where:
I is moment of inertia of solid of mass m about axis located at distance l from its center of mass
Icm is moment of inertia of the solid about axis passing thought the ceneter of mass and parallel the the previous axis
Angular momentum
where
is angular velocity of the solid
Newton's Second Law in angular form:
- for general case
- for constant moment of inertia
where
is net torque about axis of rotation associated with net external force
General case for rotation of system of particles
Resultant angular momentum vector of the system about arbitrary point C
where
and
are position vector and linear momentum vector for
i-th particle with respect to the point C (for
i = 1, 2, 3, ...,
n)
Resultant torque about point C associated with external forces
where
is external force applied at point
with respect to the point C (for
j = 1, 2, 3, ...,
k)
Newton's Second Law in angular form
Law of conservation of angular momentum of the system
If
then
about point C
Gyroscopic motion of spinning top
Here:
is angular velocity of the top about its axis
is vertical external force applied to the top
is radius-vector of the point where the force
is applied to the top
is precessional frequency of the top about z-axis
Equation of motion for the top
where I is moment of inertia of the top about it's axis
The value of precessional frequency
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