<< Chapter < Page Chapter >> Page >
  • Define arc length, rotation angle, radius of curvature and angular velocity.
  • Calculate the angular velocity of a car wheel spin.

In Kinematics , we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.

Rotation angle

When objects rotate about some axis—for example, when the CD (compact disc) in [link] rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit    used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle     Δ θ size 12{Δθ} {} to be the ratio of the arc length to the radius of curvature:

Δ θ = Δ s r . size 12{Δθ= { {Δs} over {r} } "."} {}

The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.
All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle Δ θ size 12{Δθ} {} in a time Δ t size 12{Δt} {} .

A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.
The radius of a circle is rotated through an angle Δ θ size 12{Δθ} {} . The arc length Δs size 12{Δs} {} is described on the circumference.

The arc length     Δ s size 12{Δs} {} is the distance traveled along a circular path as shown in [link] Note that r size 12{r} {} is the radius of curvature    of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius r size 12{r} {} . The circumference of a circle is r size 12{2πr} {} . Thus for one complete revolution the rotation angle is

Δ θ = r r = . size 12{Δθ= { {2πr} over {r} } =2π"."} {}

This result is the basis for defining the units used to measure rotation angles, Δ θ size 12{Δθ} {} to be radians    (rad), defined so that

rad = 1 revolution. size 12{2π" rad "=" 1 revolution."} {}

A comparison of some useful angles expressed in both degrees and radians is shown in [link] .

Comparison of angular units
Degree Measures Radian Measure
30º size 12{"30"°} {} π 6 size 12{ { {π} over {6} } } {}
60º size 12{"60"°} {} π 3 size 12{ { {π} over {3} } } {}
90º size 12{"90"°} {} π 2 size 12{ { {π} over {2} } } {}
120º size 12{"120"°} {} 3 size 12{ { {2π} over {3} } } {}
135º size 12{"135"°} {} 4 size 12{ { {3π} over {4} } } {}
180º size 12{"180"°} {} π size 12{π} {}
A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.
Points 1 and 2 rotate through the same angle ( Δ θ size 12{Δθ} {} ), but point 2 moves through a greater arc length Δ s size 12{ left (Δs right )} {} because it is at a greater distance from the center of rotation ( r ) size 12{ \( r \) } {} .

If Δ θ = 2 π size 12{Δθ=2π} {} rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are 360º size 12{"360"°} {} in a circle or one revolution, the relationship between radians and degrees is thus

2 π rad = 360º size 12{2π" rad"="360" rSup { size 8{ circ } } } {}

so that

1 rad = 360º 57. . size 12{1" rad"= { {"360" rSup { size 8{ circ } } } over {2π} } ="57" "." 3 rSup { size 8{ circ } } "."} {}

Angular velocity

How fast is an object rotating? We define angular velocity     ω size 12{ω} {} as the rate of change of an angle. In symbols, this is

ω = Δ θ Δ t , size 12{ω= { {Δθ} over {Δt} } ","} {}

where an angular rotation Δ θ size 12{Δθ} {} takes place in a time Δ t size 12{Δt} {} . The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity ω size 12{ω} {} is analogous to linear velocity v size 12{v} {} . To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length Δ s size 12{Δs} {} in a time Δ t size 12{Δt} {} , and so it has a linear velocity

Questions & Answers

how to study physic and understand
Ewa Reply
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask