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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.