question_answer

A simple pendulum with length L and mass of the bob is vibrating with an amplitude a. Then the maximum tension in the string is

A)  mg           

B)   \[mg\left[ 1+{{\left( \frac{a}{L} \right)}^{2}} \right]\]

C)  \[mg{{\left[ 1+\frac{a}{2L} \right]}^{2}}\] 

D)   mg\[{{\left[ 1+\left( \frac{a}{L} \right) \right]}^{2}}\]

Correct Answer: B

Solution :

 Tension in the string \[T-mg\cos \theta =\frac{m{{v}^{2}}}{L}\] T is maximum, when \[\theta =0{}^\circ \] \[\therefore \] \[{{T}_{\max }}=mg+\frac{mv_{\max }^{2}}{L}\] \[=mg+\frac{m{{a}^{2}}{{\omega }^{2}}}{L}\] \[\because \]a is amplitude \[=mg+\frac{m{{a}^{2}}}{L}\times \frac{g}{L}\]    \[[\because mg=m{{\omega }^{2}}L]\] \[=mg\left[ 1+{{\left( \frac{a}{L} \right)}^{2}} \right]\]