# 8.11 Unbalanced force system  (Page 3/3)

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As a matter of fact, this velocity profile is typical of any vehicle, which is first accelerated, then run with minimum acceleration or even zero acceleration and finally brought to rest with constant deceleration.

We work out an example for studying motion under variable force, which is similar to the earlier example except that force is now dependent on time.

Problem : A block of mass “m” is pulled by a string on a smooth horizontal surface with force F = kt, where “k” is a constant. The string maintains an angle “θ” with horizontal. Find the time when block breaks off from the surface.

Solution : Here, important thing is to know the meaning of "breaking off". It means that physical contact between two surfaces is broken off. In that condition, the normal force should disappear as there is no contact between block and surface.

Since normal force is directed vertically up, we need to analyze forces in y-direction only so that we could apply the condition corresponding to "breaking off".

$\text{Free body diagram of the block}$

$\begin{array}{l}\sum {F}_{y}=N+kt\mathrm{sin}\theta -mg=0\\ ⇒N=mg-kt\mathrm{sin}\theta \end{array}$

Now, let $t={t}_{B}$ (break off time) when N = 0 (breaking off condition)

$\begin{array}{l}⇒0=mg-k{t}_{B}\mathrm{sin}\theta \\ ⇒k{t}_{B}\mathrm{sin}\theta =mg\\ ⇒{t}_{B}=\frac{mg}{k\mathrm{sin}\theta }\end{array}$

## Motional mechanism of animals

According to laws of motion, motion of a body is not possible by the application of internal force. On the other hand, animals move around using internal muscular force. This is not a contradiction, but an intelligent maneuvering on the part of animals, which use internal muscular force to generate external force on them.

Let us take the case of our own movement. So long we stand upright, applying weight on the ground in vertical direction; there is no motion.

To move forward (say), we need to press back the surface at an angle. The ground applies an equal and opposite force (say, reaction of the ground). The reaction of the surface is an external force for our body. The horizontal component of the reaction force moves our body forward, whereas the vertical force balances our weight.

## Elements of body system

The underlying frame work of the analysis of force systems in inertial frame of reference is now almost complete. There is nothing new as far as application of laws of motion is concerned. But, there is a big “but” with respect to details of various elements of body systems that we consider during study of motion. These elements typically are block, string, incline, pulleys and spring.

The whole gamut of analysis in dynamics requires systemic approach to answer following questions :

• what are the forces ?
• which of them are external forces ?
• does friction is part of the external force system ?
• whether forces are collinear, coplanar or three-dimensional ?
• are forces balanced or unbalanced ?
• are the forces time dependent ?
• is the motion taking place in inertial frame or accelerated frame ?
• what would be the appropriate coordinate system for analysis ?
• what are the characteristics of system elements ?

It is not very difficult to realize that our job is half done if we are able to classify the system in hand based on the answers to above questions. Though, we have listed the system elements at the end of the list, we shall soon realize that a great deal of our effort in getting answers to questions 1, 2, 4 and 8 are largely determined by the elements involved, while the rest are situation specific.

We have briefly described system elements like block, string, pulley etc. In subsequent modules, we shall emphasize details of these and other elements.

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