Example 4.1(An Introduction to Mechanics, 2E,Kleppner and Kolenkow)
The Bola is used by gauchos for entangling their cattle. It consists of three balls of stone or iron connected by thongs. The gaucho whirls the bola in the air and hurls it at the animal. What can we say about its motion?
Why didn't the author consider centripetal force on each objects?
Equation (I) should be
$$\frac{d\vec{P}}{dt}=m_1g+m_2g+m_3g+\frac{m_1 v_1^2}{r}+\frac{m_2 v_2^2}{r} +\frac{m_3 v_3^2}{r} $$ instead of the given one? Please help me to understand the problem.
Because those are virtual forces, if you write them in a rotating reference frame. From a fixed reference frame, or one moving with a constant velocity, you need to replace those forces with $m\vec a$ instead. Another way of thinking is asking yourself what (or who) is generating the centripetal force? If I look at an object moving in a circle I can write $$ma=m\frac{v^2}r=F$$ Here $F$ might be a tension in a string or gravity for the planets. If I am in a rotating reference frame where I see the object fixed, even if I have a force $F$ acting on it, I conclude that there must be something else there as well to cancel $F$. But that force is not real, since it only exists in the rotating reference frame.