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between n, T and w.
definition, period (T) is the time taken for one
revolution to be completed. In this time, the angular displacement of
the particle is 2p radians.
Hence, angular speed w = angle traced by the radius vector in unit
\ w =
Frequency 'n' is the
number of revolutions completed in unit time.
\ n =
\ w = 2 p n.
In a clock,
For an hour hand, period T
12 × 60 × 60 seconds.
hand, period T
hand, period T
between Linear Velocity and Angular Velocity (U.C.M.)
Consider a particle performing U.C.M. in an
anticlockwise sense as shown in the figure. In a very small time
interval dt, the particle moves from the point P1
to the point P2 through the distance ds. In the
same time interval, the radius vector rotates through an angle dq.
arc p1 p2
The linear speed v of a particle is the rate at which the distance
is covered by the particle with respect to time.
Linear speed v
(r is constant)
The angular speed of a particle is the rate at which the angle is
traced by the radius vector with respect to time.
From the equation (1) and (2),
v = r w
In vector form =
between Linear Acceleration and Angular Acceleration
We have the relation,
\ v = w r. …(1)
In non-uniform circular motion, w is not constant and the particle has a
linear acceleration (aT) in the tangential direction.
\ aT = µ t.
aT = r a
a = aR
in Uniform Circular Motion
Consider a particle moving around
a circular path of radius r having O as its centre. Let its angular
speed be w which is constant.
In time t, let the
particle move from A to P.
The angle covered in this time = Ð POM = q = wt.
At time t, the radius vector
cos w t +
r sin w t
w sin w t +
r w cos w t
From (2), v = r w
w 2 cos w t -
r w 2 sin
- w 2
The magnitude of the acceleration is a = w 2
r and the
negative sign indicates that the direction of the acceleration is
opposite to that of the radius vector . The
acceleration is directed along the radius towards the centre of the
circle. This acceleration is known as the centripetal
("centre - seeking") or radial acceleration.