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An object executing a circular orbit of radius with uniform tangential
speed possesses a velocity vector whose magnitude is constant, but
whose direction is continuously changing. It follows that the object must
be accelerating, since (vector) acceleration is the rate of change of (vector)
velocity, and the (vector) velocity is indeed varying in time.
Figure 58:
Centripetal acceleration.

Suppose that the object moves from point to point between times and , as
shown in Fig. 58. Suppose, further, that the object rotates
through radians in this time interval. The vector
,
shown in the diagram, is identical
to the vector
. Moreover, the angle subtended between vectors
and
is simply . The vector
represents
the change in vector velocity, , between times and . It can be seen
that this vector is directed towards the centre of the circle. From standard trigonometry,
the length of vector
is

(256) 
However, for small angles
, provided that
is measured in radians. Hence,

(257) 
It follows that

(258) 
where
is the angular velocity of the object, measured
in radians per second.
In summary, an object executing a circular orbit, radius , with uniform tangential
velocity , and uniform angular velocity , possesses an acceleration
directed towards the centre of the circlei.e., a centripetal accelerationof
magnitude

(259) 
Figure 59:
Weight on the end of a cable.

Suppose that a weight, of mass , is attached to the end of a cable, of
length , and whirled around such that the weight executes a horizontal circle,
radius , with uniform tangential velocity . As we have just learned,
the weight is subject to a centripetal acceleration of magnitude . Hence,
the weight experiences a centripetal force

(260) 
What provides this force? Well, in the present example, the force is provided by the
tension in the cable. Hence, .
Suppose that the cable is such that it snaps whenever the tension in it
exceeds a certain critical value . It follows that there is a
maximum velocity with which the weight can be whirled around: namely,

(261) 
If exceeds then the cable will break. As soon as the cable snaps,
the weight will cease to be subject to a centripetal force, so it will fly offwith
velocity along the straightline
which is tangential to the circular orbit it was previously executing.
Next: The conical pendulum
Up: Circular motion
Previous: Uniform circular motion
Richard Fitzpatrick
20060202