Page 95 - Class-11-Physics-Part-1_Neat

MOTION IN A PLANE 81

 v  2 2 2
a =   v = v /R (4.44) v ω R 2
2
c  R  a = = = ω R
c
R R
Thus, the acceleration of an object moving with 2
a c = ω R (4.47)
speed v in a circle of radius R has a magnitude
2
v /R and is always directed towards the centre.
This is why this acceleration is called centripetal The time taken by an object to make one revolution
acceleration (a term proposed by Newton). A is known as its time period T and the number of
thorough analysis of centripetal acceleration was revolution made in one second is called its
first published in 1673 by the Dutch scientist frequency ν (=1/T). However, during this time the
Christiaan Huygens (1629-1695) but it was distance moved by the object is s = 2πR.
probably known to Newton also some years earlier. Therefore, v = 2πR/T =2πRν (4.48)
“Centripetal” comes from a Greek term which means In terms of frequency ν, we have
‘centre-seeking’. Since v and R are constant, the ω = 2πν
magnitude of the centripetal acceleration is also v = 2πRν
constant. However, the direction changes — 2 2
pointing always towards the centre. Therefore, a a = 4π ν R (4.49)
c
centripetal acceleration is not a constant vector. t Example 4.10 An insect trapped in a
We have another way of describing the
velocity and the acceleration of an object in circular groove of radius 12 cm moves along
the groove steadily and completes 7
uniform circular motion. As the object moves
from P to P′ in time ∆t (= t′ – t), the line CP revolutions in 100 s. (a) What is the
(Fig. 4.19) turns through an angle ∆θ as shown angular speed, and the linear speed of the
motion? (b) Is the acceleration vector a
in the figure. ∆θ is called angular distance. We constant vector ? What is its magnitude ?
define the angular speed ω (Greek letter omega)
as the time rate of change of angular Answer This is an example of uniform circular
displacement :
motion. Here R = 12 cm. The angular speed ω is
∆ θ
ω = given by
∆t (4.45) ω = 2π/T = 2π × 7/100 = 0.44 rad/s
The linear speed v is :
Now, if the distance travelled by the object
-1
during the time ∆t is ∆s, i.e. PP′ is ∆s, then : v =ω R = 0.44 s × 12 cm = 5.3 cm s -1
∆ s The direction of velocity v is along the tangent
v =
t ∆ to the circle at every point. The acceleration is
directed towards the centre of the circle. Since
but ∆s = R ∆θ. Therefore :
this direction changes continuously,
∆θ
v = R = R ω acceleration here is not a constant vector.
t ∆ However, the magnitude of acceleration is
v = R ω (4.46) constant:
2
–1 2
We can express centripetal acceleration a in a = ω R = (0.44 s ) (12 cm)
c
terms of angular speed : = 2.3 cm s -2 t

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