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Page # 6
Fext
ROTATIONAL DYNAMICS
m1a1 m 2 a2
m3 a 3 .............
Where m1, m2, m3, ......... are the masses of different particles of the body having accelerations
a1, a2 , a3 ,............... respectively..
But acceleration of all the particles are same So, a1
Fext
a2
a3
.........
a
Ma
Where M = Total mass of the body
a = acceleration of any particle or of centre of mass of body
P m1v1 m2 v 2 m3 v 3 .............
Where m1, m2, m3 ...... are the masses of different particles of the body having velocities
v1, v2 , v3 ............. respectively
But velocities of all the particles are same so v1
v2
v 3 .......... v
P Mv
Where v = velocity of any particle or of centre of mass of the body..
Total Kinetic Energy of body =
1
m1v12
2
1
m2 v 22
2
.......... .
1
Mv 2
2
1.2. Pure Rotational Motion :
A body is said to be in pure rotational motion if the perpendicular distance of each particle
remains constant from a fixed line or point and do not move parallel to the line, and that line
is known as axis of rotation. In this motion all the particles have same ,
instant. Eg. : - a rotating ceiling fan, arms of a clock.
For pure rotation motion :s
Where = angle rotated by the particle
r
s = length of arc traced by the particle.
r = distance of particle from the axis of rotation.
m2
= angular speed of the body..
at an
m1
m3
d
Where
dt
and
m4
m5
m6
d
Where = angular acceleration of the body..
dt
All the parameters , and are same for all the particles. Axis of rotation is perpendicular to
the plane of rotation of particles.
Special case : If = constant,
= 0+ t
Where 0 = initial angular speed
1 2
t t = time interval
2
2
= 02 + 2
0t
Total Kinetic Energy
1
m1v12
2
1
[m1r12
2
1
m2 v 22 .................
2
m2 r22 ................]
2
1 2
I
Where I = Moment of Inertia = m1r12
2
= angular speed of body.
m2r22 .......
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