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A uniform rope of mass M and length L is fixed to its upper end vertically from a rigid support. Thus the tension in the rope at the distance I from the rigid support is
\[
  {\text{A}}{\text{. Mg}}\dfrac{{\text{L}}}{{{\text{L + I}}}} \\
  {\text{B}}{\text{. }}\dfrac{{{\text{Mg}}}}{{\text{L}}}({\text{L - I)}} \\
  {\text{C}}{\text{. Mg}} \\
  {\text{D}}{\text{. }}\dfrac{{\text{I}}}{{\text{L}}}{\text{Mg}} \\
\]

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Last updated date: 20th Jun 2024
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Answer
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Hint : Tension is described as the pulling force transmitted axially by the means of a string, a cable, chain, or by each end of a rod.

Complete step by step solution:
Mass = M
Length = L
Tension = mass × acceleration due to gravity
T = mg
Now we have to do the differentiating we get
dT = dmg
\[{\text{dm = }}\dfrac{{{\text{Mdx}}}}{{\text{L}}}\]
Integrating from I to L
\[{\text{T = }}\dfrac{{{\text{Mdxg}}}}{{\text{I}}} = \dfrac{{{\text{Mg(L - I)}}}}{{\text{L}}}\] \[\]
The ends of a string or the other object transmitting tension will exert forces on the object to which the string or the rod is connected in the direction of the string at the point of attachment.

Thus option B is the correct answer.

Additional information:
String-like objects in relativistic theories, such as the strings used in some models of the instructions between quarks, or those used in the modern theories also possess tension. These strings are analysed in terms of their word sheet, and the energy is then typically proportional to the length of the length of the strings. Tension in a string is a scalar quantity. Zero tension is slack. A string or rope is often idealised as one dimension, having length but being, mass less with zero cross section. Tension is also used to describe the force exerted by the ends of the dimensional, continuous materials such as rod or turns member.

Note:
There are the two basic possibilities for the system of the objects held by the strings. Either acceleration is zero and the system is therefore in equilibrium, or there is acceleration, and therefore a net force is present in the system. Tensions are the transmitted force as an action reaction pair of the forces, or as a restoring force may be a force and has the units of force measured in Newton.