*Note: Type-setting limitations do not allow for arrows on the vector labels, so we have used boldface only.Concepts and Skills to Review
Gravitational forces (Section 2.5 )
Newton's second law: force and acceleration (Section 3.4 )
Velocity and acceleration (Section 4.4 )
Summary
The angular displacement Δθ is the angle through which an object has turned. Positive and negative angular displacements indicate rotation in different directions. Conventionally, positive represents counterclockwise motion.
The instantaneous angular velocity and acceleration are the limits of the average quantities as <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0070524076/57988/image5_a.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a>
A useful measure of angle is the radian:
2π radians = 360°
Using radian measure for θ, the arc length s of a circle of radius r subtended by an angle θ is:
s = θr (θ in radian measure)
(5-5)
Using radian measure for ω, the speed of an object in circular motion (including a point on a rotating object) is
v = rω (ω in radians per unit time)
(5-6)
Using radian measure for α, the tangential acceleration component is related to the angular acceleration as follows:
at= rα (α in radians per time2)
(5-17)
An object moving in a circle has a centripetal acceleration component given by
The tangential and centripetal accelerations are two perpendicular components of the acceleration. The centripetal acceleration component changes the direction of the velocity and the tangential acceleration component changes the speed.
Uniform circular motion means that v and ω are constant. In uniform circular motion, the time to complete one revolution is constant and is called the period T. The frequency f is the number of revolutions completed per second.
where the SI unit of angular velocity is rad/s and that of frequency is rev/s = Hz.
A rolling object is both rotating and translating. If the object rolls without skidding or slipping, then
vaxle = rω
Kepler's third law says that the square of the period of a planetary orbit is proportional to the cube of the orbital radius:
T2 = constant × r3
(5-14)
In the case of constant angular acceleration, we can solve rotational kinematics problems using relationships analogous to those we developed for linear motion:
