When Galileo put forward the proposition that the motion of a projectile analyzed into separate horizontal & vertical parts, he made a great contribution to the conceptual basis of mechanics. It may be worth pointing out , however, that this independence of motions breaks down if one takes into account the resistive force exerted on objects of ordinary size.

Consider an object moving in a vertical plane. At an instant, let its velocity $\mathbf{v}$ be directed at $\theta$. The object is then subjected gravitional force & resistive force of magnitude $Bv^2$ opposite to $\mathbf{v}$. Newton's law applied to the x & y components of motion at that instant, thus gives us: $$m\dfrac{dv_x}{dt} = - (Bv) . v_x \quad \& \quad m\dfrac{dv_y}{dt} = -mg -(Bv) . v_y$$. Thus the equation governing each separate component of velocity involves the knowledge of the magnitude of the total velocity and hence of what is happening at each instant along the other coordinate direction. The larger the magnitude of $\mathbf{v}$, the more important does this cross connection between the different components of the motion become. We cannot calculate the vertical motion of the body without reference to the horizontal component. -A.P.French's Newtonian Mechanics.

So, only the presence of $v$ make the equations dependent? And does dependence mean we cannot analyze the motion using components like in the general projectile motion?

Now, if that is so, does that mean the x-component of centripetal force by tension on a peg in uniform circular motion is dependent on y- component ie: $$F_x = -T\cos\theta = m\dfrac{v^2}{A} \cos\theta = \dfrac{mv}{A} v_y$$, where $\theta$ is the angle made by $A$ with the horizontal? That means SHM in x- axis is dependent on that of y-axis. Really?

Actually what does dependence mean?

  • $\begingroup$ 2D quadratic drag was also considered in this Phys.SE post and links therein. $\endgroup$
    – Qmechanic
    Apr 22 '15 at 18:29

Yes, only the presence of $v = \sqrt{v_x^2 + v_y^2}$ makes these equations mathematically dependent. We can continue to analyze the equations using components (as is in fact being done here), but they will no longer separate out neatly like in problems of projectile motion.

Regarding the case of circular motion, you can see that there must be an interdependence between the $x$ and $y$ components i.e. SHM in these two directions must be related. This is indirectly an expression of the constraint $x^2 + y^2 = A^2$, for a circle of radius $A$. Complete independence of these components would mean that you could achieve any kind of motion (say eccentric ellipses), as opposed to that constrained to a circle: to see this, consider the case when the $x$ and $y$ SHMs are in phase, giving a straight line instead of a circle.

The situation gets more complicated in other coordinate systems, where there is interdependence of components in the equations of motion themselves. For example, in polar coordinates, the radial force does not directly lead to a radial acceleration: the centrifugal force, which depends on angular velocity, also plays a role.

In general, most physical problems have such relations between components of various quantities, and therefore one would encounter such equations more often than those of the simple, separable kind.

  • $\begingroup$ Sir, I've modified the question greatly adding the source and my confusion, queries. Please read it. And request you to please help. $\endgroup$
    – user36790
    Apr 22 '15 at 18:45
  • $\begingroup$ Thanks for your help. Really a good answer. However, I do want to know, how to analyze one equation using the other. And Mr. French wrote that these type of equations are solved by numerical-methods - methods of digital computer. Now, what method is he talking about? $\endgroup$
    – user36790
    Apr 23 '15 at 9:59
  • $\begingroup$ Can you please quantify your example of polar coordinate? Does SHM occur only in circle? What about that you said like eccentric ellipse? What would happen then? Can you elaborate please? BTW, if the components are in phase, isn't the locus a circle of radius unity? $\endgroup$
    – user36790
    Apr 23 '15 at 10:30
  • $\begingroup$ I'm not aware of those numerical methods - except for going step-by-step with approximations. SHM along the axes also occurs for elliptical motions. If the components are in phase, the overall path is a line. Only when they are 90 degrees out of phase is it a circle (or ellipse with major and minor axes along the coordinate axes). As for polar coordinates, see: en.wikipedia.org/wiki/Polar_coordinate_system $\endgroup$
    – AV23
    Apr 23 '15 at 10:44

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