Jee main notes for rotational motion, rigid body dyanamics

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,ROTATIONAL
MECHANICS

, MARKS 3

ROT
OTAATIONAL ME
MECCHANI
NICCS
Linear Motion Rotational Motion
Position x  Angular position
Velocity v  Angular velocity
Acceleration a  Angular acceleration
Motion equations x = v t  t
=  Motion equations
v = v0 + at  = 0 + t

1 2 1 2
x = v0t  at   0 t  t
2 2

v2 = v02  2ax 2 = 20  2

Mass (linear inertia) m I Moment of inertia
NewtonÊs second law F = ma  = I 
Momentum p = mv L = I  Angular momentum
Work Fd  Work

1 1 2
Kinetic energy mv2 I Kinetic energy
2 2

Power Fv  Power

1. INTR ODUC TION
Consider a disk rotating on a stationary rod.




If we now view this disk from the top,




S

r




ROTATIONAL MECHANICS

, 4 MARKS
we see that when the disk rotates so that the arc length (s) equals the length of the radius of
the disk (r ), the subtended central angle () will equal 1 radian. The resulting equation is

s = r
or
 = s/r

where the unit of a radian represents the dimensionless measure of the ratio of the circleÊs arc
length to its radius.

If the disk rotates through one complete revolution, then s equals the entire circumference and
 equals 2 radians.

s = r
2r = r
 = 2 radians

Differentiating the basic equation s = r results in the next two relationships between the tangential
motion of a point along the circumference and the angular behavior of the disk itself.

s = r
ds/dt = r (d /dt)
v = r

The Greek letter  represents the angular velocity of the disk. It is measured in the unit
radians/sec. Differentiating one more time, we have

v = r
dv/dt = r (d /dt)
a = r

The Greek letter  represents the angular acceleration of the disk. It is measured in the unit
radian/sec2.

Summ a r y

These three equations

s = r
v = r
a = r

allow us to relate the linear motion (s, v, a) of a point moving in circular motion on a rotating
platform with the the rotational motion (, , ) of the platform itself.



ROTATIONAL MECHANICS