angular momentum:
change in angular velocity:
kinematics of rotational motion:
law of conservation of angular momentum:
moment of inertia:
right-hand rule:
rotational inertia:
rotational kinetic energy:
tangential acceleration:
torque:
work-energy theorem:
the product of moment of inertia and angular velocity
the difference between final and initial values of angular velocity
describes the relationships among rotation angle, angular velocity, angular acceleration, and time
angular momentum is conserved, i.e., the initial angular momentum is equal to the final angular
momentum when no external torque is applied to the system
mass times the square of perpendicular distance from the rotation axis; for a point mass, it isI=mr^2 and, because any
object can be built up from a collection of point masses, this relationship is the basis for all other moments of inertia
direction of angular velocity ω and angular momentum L in which the thumb of your right hand points when you curl your fingers
in the direction of the disk’s rotation
resistance to change of rotation. The more rotational inertia an object has, the harder it is to rotate
the kinetic energy due to the rotation of an object. This is part of its total kinetic energy
the acceleration in a direction tangent to the circle at the point of interest in circular motion
the turning effectiveness of a force
if one or more external forces act upon a rigid object, causing its kinetic energy to change fromKE 1 toKE 2 , then the
workW done by the net force is equal to the change in kinetic energy
Section Summary
10.1 Angular Acceleration
• Uniform circular motion is the motion with a constant angular velocityω=Δθ
Δt
.
• In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) isα=Δω
Δt
.
• Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given asat=Δv
Δt
.
• For circular motion, note thatv=rω, so that
at=
Δ(rω)
Δt
.
• The radius r is constant for circular motion, and soΔ(rω)=rΔω. Thus,
at=rΔω
Δt
.
• By definition,Δω/ Δt=α. Thus,
at=rα
or
α=
at
r.
10.2 Kinematics of Rotational Motion
- Kinematics is the description of motion.
- The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
- Starting with the four kinematic equations we developed in theOne-Dimensional Kinematics, we can derive the four rotational kinematic
equations (presented together with their translational counterparts) seen inTable 10.2.
• In these equations, the subscript 0 denotes initial values (x 0 andt 0 are initial values), and the average angular velocityω- and average
velocity v- are defined as follows:
ω ̄ =
ω 0 +ω
2
and v ̄ =
v 0 +v
2
.
10.3 Dynamics of Rotational Motion: Rotational Inertia
- The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
• If we exert a forceFon a point massmthat is at a distancerfrom a pivot point and because the force is perpendicular tor, an
accelerationa = F/mis obtained in the direction ofF. We can rearrange this equation such that
F = ma,
and then look for ways to relate this expression to expressions for rotational quantities. We note thata = rα, and we substitute this expression
intoF=ma, yielding
F=mrα
CHAPTER 10 | ROTATIONAL MOTION AND ANGULAR MOMENTUM 349
