The elements of a body system connected by a string have common acceleration. Consider the body system comprising of two blocks. As the system comprises of inextensible elements, the body system and each element, constituting it, have same magnitude of acceleration (not acceleration as there is change of direction), given by :

Connected system involving string |
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In the process, string transmits force undiminished for the ideal condition of being “inextensible” and “mass-less”. This is the basic character of string. We must however be careful in stating what we have stated here. If the ideal condition changes, then behavior of string will change. If string has certain “mass”, then force will not be communicated “undiminished”. If string is extensible, then it behaves like an extensible spring and accelerations of the string will not be same everywhere.

** Inextensible and mass-less string
**

Determination of common acceleration and tension in the string is based on application of force analysis in component forms. In general, force analysis for each of the block gives us an independent algebraic relation. It means that we shall be able to find as many unknowns with the help of as many equations as there are blocks in the body system.

Individual string has single magnitude of tension through out its length. On the other hand, if the systems have different pieces of strings (when there are more than two blocks), then each piece will have different tensions.

#### Example 1

Problem :
Two blocks of masses “

Connected system involving string |
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Solution :
Since there is no friction, the block of mass "

Two blocks are connected by a taut string. Hence, magnitude of accelerations of the bodies are same. Let us assume that the system moves with magnitude of acceleration “a” in the directions as shown. It should be understood here that a string has same magnitude of acceleration - not the acceleration as string in conjunction with pulley changes the direction of motion and hence that of acceleration.

The external forces on "

On the other hand, the external forces on "

Combining two equations, we have :

and

** Useful deduction
**

The result for acceleration as obtained above is significant for this type of arrangement, where blocks are pulled down by a string passing over a pulley. The acceleration as derive is equivalent to :

where

Connected system involving string |
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In this case,

Important to note here is that the separate strings have different tensions.

** Mode of force application
**

Consider the two cases when a block is pulled down by a force in two different manners. In one case, string is pulled by another mass

Connected system involving string |
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As worked out earlier, the acceleration in the first case,

In the second case, let the acceleration of the block be “

But,

Now, combining two equations, we have :

Evidently,

** Variable acceleration of a single string
**

Further, the behavior of string is true to ideal string so long string is not intervened by other element, which is itself accelerating. Consider a pulley – string system, in which the pulley is accelerating as shown in the figure.

String in contact with moving pulley |
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In such case, the accelerations of different parts of the string are not same. Note in the example shown above that one end of the string is fixed, but the other end of string is moving. As a matter of fact, the acceleration of block (hence that of the string at that end) is twice that of the pulley. The acceleration of the other end of the string, which is fixed, is zero. This aspect is explained in a separate module on movable pulley in the course.

** String with certain mass
**

The tension in the string having certain mass is not same everywhere. A part of the force is required to accelerate string as well. Usually,mass is distributed uniformly along the length of the string. We describe mass distribution in terms of "mass per unit length" and denotes the same as "λ" such that the mass of the string,"m" of length "L" is given as :

m = λL

If we take cross section at any point in the string, then the part of the string on each side forms an mass element. having mass proportional to the length of that part of the string. This aspect is brought out in the example given here.

#### Example 2

Problem : A body of mass “m” is hanging with a string having linear mass density “λ”. What is the tension at point “A” as shown in the figure.

Balanced force system |
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Solution : The tension in the string is not same. The tension above “A” balances the weight of the block and the weight of the string below point “A”.

Free body diagram |
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Mass of the string below point A, "

The external forces at point "A" are (i) Tension, T (ii) weight of block, mg, and (iii) weight of string below A,"