PHYS 103 (GENERAL PHYSICS)

CHAPTER 10: Rotation of A Rigid Object about A

Fixed Axis

King Saud University

College of Science

Physics & Astronomy Dept.

Presented by Nouf Saad Alkathran

نوف الخضران. أ 2

A particle at P is at a fixed distance r from the origin and rotates in a circle

of radius r. It is convenient to represent the position of P with its polar

coordinates (r, Ɵ) where r is the distance from the origin to P and Ɵ is

measured counterclockwise from some reference line.

The arc length s is related to the angle

Ɵ through the relationship

Ɵ unit is radian (rad)

1 revolution =360(degree) =2π (rad) : 1 rad ≈57.3◦

Angular displacement ∆𝜃:

Average angular speed ω :

Instantaneous angular speed ω:

Angular speed has units of radians per second

(rad/s), which can be written as

because radians are not dimensional.

نوف الخضران. أ 3

Average angular acceleration 𝛼 :

Instantaneous angular acceleration:

نوف الخضران. أ 4

When a rigid object is rotating about a fixed axis,

every particle on the object rotates through the same

angle in a given time interval and has the same angular

speed and the same angular acceleration. That is, the

quantities Ɵ,ω, and α, characterize the rotational motion of

the entire rigid object as well as individual particles in the

object.

To illustrate the direction of ω and α, it is convenient to

use the right-hand rule demonstrated. When the four fingers

of the right hand are wrapped in the direction of rotation Ɵ,

the extended right thumb points in the direction of ω. The

direction of α follows from its definition α=dω/dt. It is in the

same direction as if the angular speed is increasing in time,

and it is antiparallel to ω if the angular speed is decreasing in

time.

نوف الخضران. أ 5

نوف الخضران. أ 6

They can be generated from the equations for linear

motion by making the substitutions

نوف الخضران. أ 7

(B) Through how many revolutions has the wheel turned during this time

interval?

(C) What is the angular speed of the wheel at t " 2.00 s?

نوف الخضران. أ 8

Tangential velocity and angular speed:

Tangential acceleration and angular acceleration:

The total linear acceleration vector:

نوف الخضران. أ 9

نوف الخضران. أ 10

kinetic energy determined by its mass

and tangential speed. If the mass of the

particle is mi and its tangential speed is vi, its

kinetic energy is

The total kinetic energy of the rotating rigid

object is the sum of the kinetic energies of

the individual particles:

نوف الخضران. أ 11

Four tiny spheres are fastened to the ends of two rods of negligible

mass lying in the xy plane. We shall assume that the radii of the spheres

are small compared with the dimensions of the rods.

(A) If the system rotates about the y axis with aangular speed find the

moment n of inertia and the rotational kinetic energy about this axis.

نوف الخضران. أ 12

(B) Suppose the system rotates in the xy plane

about an axis (the z axis) through O .

Calculate the moment of inertia and

rotational kinetic energy about this axis.