Consider Fig. 64. Suppose that the object moves from point , where its tangential
velocity is , to
point , where its tangential velocity is . Let us, first of all, obtain the relationship
between and . This is most easily achieved by considering energy conservation.
At point , the object is situated a vertical distance below the pivot, whereas at point the
vertical distance below the pivot has been reduced to . Hence, in moving
from to the object gains potential energy
. This gain
in potential energy must be offset by a corresponding loss in kinetic energy. Thus,

(290) |

Let us now examine the radial acceleration of the object at point . The radial forces
acting on the object are the tension in the rod, or string, which acts towards
the centre of the circle, and the component
of the object's weight,
which acts away from the centre of the circle. Since the object is executing
circular motion with instantaneous tangential velocity , it must experience
an instantaneous acceleration towards the centre of the circle. Hence,
Newton's second law of motion yields

Suppose that the object is, in fact, attached to the end of a piece of string, rather than a
rigid rod. One important property of strings is that, unlike rigid rods, they cannot
support negative tensions. In other words, a string can only pull objects attached to its
two ends together--it cannot push them apart. Another way of putting this is that if the
tension in a string ever becomes negative then the string will become slack and collapse.
Clearly, if our object is to execute a full vertical circle then then tension in the
string must
remain positive for all values of . It is clear from Eq. (293) that the
tension attains its minimum value when
(at which point ). This
is hardly surprising, since
corresponds to the point at which the
object attains its maximum height, and, therefore, its minimum tangential velocity. It is certainly
the case that if the string tension is positive at this point then it must be positive
at all other points. Now, the tension at
is given by

(294) |

Note that this condition is independent of the mass of the object.

Suppose that the object is attached to the end of a rigid rod, instead of a
piece of string. There is now no constraint on the tension, since a rigid rod can
quite easily support a negative tension (*i.e.*, it can push, as well as pull, on objects
attached to its two ends).
However, in order for the object to execute a complete vertical circle the square of its
tangential velocity must remain positive at all values of .
It is clear from Eq. (291) that
attains its minimum value when
. This
is, again, hardly surprising. Thus, if is positive at this point then it must be positive
at all other points. Now, the expression for at
is

(296) |

(297) |