MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Circular Motion Page 2

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Relation between n, T and w.

By 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 time.

 \ w = 2 p T

Frequency '
n' is the number of revolutions completed in unit time.

 \ n = 1 T

\ w = 2 p n.
In a clock,

 For an hour hand, period T = 12 hours = 12 × 60 × 60 seconds. = 43,200 seconds. Minute hand, period T = 1 minute = 60 seconds. and second hand, period T = 1 second.

Relation 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.

 dq = arc p1 p2 = ds radius r

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

=

 ds dt

=

 ds dt

=

 r q dt

=

 r dq (r is constant) dt

\ v

= r

 dq ...(1) dt

The angular speed of a particle is the rate at which the angle is traced by the radius vector with respect to time.
Thus,

 w = d q ...(2) dt

From the equation (1) and (2),
v = r w
In vector form =
´

Relation 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.

Differentiating (1),

 dv = dw . r dt dt

\ aT = µ t.

Circular Motion

Uniform Circular Motion

aT = r a

 a = dw = 0 dt

a =

a = aR

Acceleration in Uniform Circular Motion
Calculus Method

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 ® = ® ® r i x + j y

 \ ® = ® ® r i r cos w t + j r sin w t ...(1)

 \ ® = ® ® v -i r w  sin w t + j r w  cos w t ...(2)

 From (2), v = r w ...(3)

\ acceleration

®

=

 = dw . r dt dt

®

®

a

-i

r

w 2 cos w t -

j

r w 2 sin w t

 ® ® s \ a = - w 2 r (4)

 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.

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