91Ó°ÊÓ

Tension is defined as the force developed is a cable, rope or string when stretched under an applied force.

Centripetal force is defined as a force that makes a body follow a curved path.

It is given by,

${\mathit{F}}_{\mathbf{cent}}\mathbf{=}\frac{{\mathbf{mv}}^{\mathbf{2}}}{\mathbf{L}}$

And for circular motion,

${\mathit{F}}_{\mathbf{cent}}\mathbf{=}\mathit{T}\mathbf{-}\mathit{m}\mathit{g}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{θ}$

Here,

${\mathit{F}}_{\mathbf{c}\mathbf{e}\mathbf{n}\mathbf{t}}$ is the centripetal force

$\mathit{L}$is the length of the string

$\mathit{θ}$is the angle with the vertical

$\mathit{m}$is the mass of the object on which the force is acting

localid="1663826566662" $\mathit{v}$is the tangential velocity of the object

localid="1663826570601" $\mathit{T}$is the tension

As shown in the figure, the tension in the string pulls the bob towards the centre, so it can only provide centripetal force to the bob which helps in the circular motion of the bob.

Using the formulas of centripetal force we get,

$$\begin{gathered} \frac{{\mathbf{mv}}^{\mathbf{2}}}{\mathbf{L}}\mathbf{=}\mathit{T}\mathbf{-}\mathit{m}\mathit{g}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{θ} \\ \mathbf{⇒}\mathit{T}\mathbf{=}\frac{{\mathbf{mv}}^{\mathbf{2}}}{\mathbf{L}}\mathbf{+}\mathit{m}\mathit{g}\mathbf{}\mathbf{cos}\mathit{θ} \end{gathered}$$

$$\begin{gathered} {\mathit{T}}_{\mathbf{Min}}\mathbf{=}\frac{\mathbf{m}{\mathbf{0}}^{\mathbf{2}}}{\mathbf{L}}\mathbf{+}\mathit{m}\mathit{g}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{θ} \\ {\mathit{T}}_{\mathbf{Min}}\mathbf{=}\mathit{m}\mathit{g}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{θ} \end{gathered}$$

Therefore, String tension will maximum at equilibrium point i.e. $\mathit{θ}\mathbf{=}\mathbf{0}$and will be least at end point. (Maximum displacement)