# A small bob tied at one end of a thin string of length

Question:

A small bob tied at one end of a thin string of length $1 \mathrm{~m}$ is describing a vertical circle so that the maximum and minimum tension in the string are in the rato $5: 1$. The velocity of the bob at the highest position is $\mathrm{m} / \mathrm{s}$. (take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

Solution:

(5)

by conservation of energy, $v_{\min }^{2}=V^{2}-4 g \mid$

$\mathrm{T}_{\max }=\mathrm{mg}+\frac{\mathrm{mv}^{2}}{\mathrm{I}} \quad \ldots(2)$

$\mathrm{T}_{\min }=\frac{\mathrm{ms}_{\min }^{2}}{1}-\mathrm{mg} \quad \ldots(3)$

from equation (1) and (3)

$T_{\min }=\frac{m}{I}\left(v^{2}-4 g l\right)-m g$

$\frac{T_{\text {max }}}{T_{\text {ain }}}=\frac{\frac{2^{2}}{1}+g}{\frac{v^{2}}{2}-5 g}$

$\frac{5}{1}=\frac{\frac{v^{2}}{T}+10}{\frac{v^{2}}{1}-50}$

$5 v^{2}-250=v^{2}+10$

$\mathrm{v}^{2}=65 \quad \ldots(4)$

from equation (4) and (1) $\mathrm{v}^{2} \mathrm{~min}=65-40=25$

$v_{\min }=5$