Conservation of energy in swinging of pendulum?

Sat, May 14, 2005 at 3:03 PM

This is yet another discovery I made in the process of making a
G.E.G. All copyright reserved by me.
We all learn from the physics textbook that energy is conserved in
any situation. An excellent example would be a falling object: At any
moment of its descending, the sum total of its kinetic energy and
potential energy should be equal. This does not bother my intellect at
all. What I found interesting is the energy exchange of the case of
swinging or rotational motion. Does they obtain the same law of
conservation of energy as the case of falling object?
The reason I think why falling object is chosen for textbook
example is because of the linearity of the motion. During the whole
trajectory, the velocity remain on the same straight line as the
acceleration as well as the line which we calculate the level of
potential energy. Now consider the case of swinging object. As we
apply a pulse force to distrub the equilibrium, what the object
respond is by displacement in two dimension. Due to the restraint of a
rope, the object must SWING instead of moving in a plane parallel to
the ground. It is easy to see that as the object is swinging, it should
have its own set of angular momentum, angular velocity and rotational
energy. Although the conventional calculation require us to calculate
the velocity at different height, in what sense does these velocity
meant? Since the object is moving in a circular trajectory, it follows
that the object is changing its direction of movement in every
moment(with centipendal acceleration). So if that velocity have any
meaning, it should refer to the instantaneous velocity in the direction
of its movement. However, as far as I understand the teaching of
Physics, it doesn’t appear that velocity refer to the velocity at the
direction of its movement. It is most situably refer to the direction
of the vertical component of the movement since it is linear.(How
meaningful/useful is this calculation is another matter.) Moreover,
any conservation of energy formula would require a condition that NO
external force is acting on the system. This condition is VIOLATED
since a varying tension of the rope is CONTINUOUSLY acting on the
system to neutralize/reduce or even overcome the gravity. It would be
a better analogy to consider this system as an object is continuously
acting by a fixed and a variable force in the opposition direction.
Now, taken these considerations into account. The following
conclusion is of a closer description of reality: At any moment, the
sum of Rotational Kinetic Energy, Work done by the centripetal force
and gravitational Potential energy should be a constant. We can apply
this formula into calculate the exact Angular Velocity of the object,
then use sine and cosine of its value to obtain the vertical and
horizontal velocity of this object.



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