Question

A particle of mass $m$ is connected with two inextensible strings $A O$ and $B O$ of equal a lengths $l$ . These two strings are finally attached to a vertical rod at points $A$ and $B$ as shown in figure. Distance between $A$ and $B$ is also $l$ . The setup is rotated with constant angular speed $ω$ with rod as the axis.

(a) Find the values of $ω$ for which the particle remains at point $B$ .

(b) Find the range of values of $ω$ for which tension $(T_{1})$ in the string $A O$ is greater than $m g$ but the other string remains slack.

(C) Find the value of $ω$ for which tension $(T_{1})$ in string $A O$ is twice the tension $(T_{2})$ in the string $B O$ .

Accepted Answer

(a) Consider a conical pendulum with one string $(A O)$ only.

$r = l \sin (θ) \dots (i) T \sin (θ) = m ω^{2} r T \sin (θ) = m ω^{2} l \sin (θ) \Rightarrow T = m ω^{2} l \dots (ii)$

and $T \cos (θ) = m g$

$\Rightarrow \cos (θ) = \frac{m g}{m ω^{2} l} = \frac{g}{ω^{2} l} \dots (iii)$

For a conical pendulum to be possible, $θ > 0^{o}$

$\Rightarrow \cos (θ) < 1 \Rightarrow \frac{g}{ω^{2} l} < 1 \sqrt{\frac{g}{l}} < ω$

Hence, particle will stay at $B$ if $ω \leq \sqrt{\frac{g}{l}}$ .

(b)

When $ω > \sqrt{\frac{g}{l}}$ , the particle begins to rotate in a circle and the tension in string becomes greater than $m g$ . The second string will get taut only when $θ = 60 °$

$\Rightarrow \cos θ = \frac{m g}{m ω^{2} l} = \frac{g}{ω^{2} l}$

$ω^{2} = \frac{g}{l \cos (60 °)} = \frac{2 g}{l} ω = \sqrt{\frac{2 g}{l}}$

If $ω > \sqrt{\frac{2 g}{l}}$ , there is tension in $B O$

Required range is $\sqrt{\frac{g}{l}} < ω \leq \sqrt{\frac{2 g}{l}}$

(c)

$T_{1} \cos (60 °) = T_{2} \cos (60 °) + m g T_{1} - T_{2} = 2 m g \dots (iv)$

and

$(T_{1} + T_{2}) \sin (60 °) = m ω^{2} r \frac{\sqrt{3}}{2} (T_{1} + T_{2}) = m ω^{2} \frac{\sqrt{3}}{2} l \Rightarrow T_{1} + T_{2} = m ω^{2} l \dots (v)$

For, $T_{1} = 2 T_{2}$ , equations $(iv)$ and $(v)$ give $T_{2} = m g$ and $3 T_{2} = m ω^{2} l$

$\Rightarrow 6 m g = m ω^{2} l \Rightarrow ω = \sqrt{\frac{6 g}{l}}$

Important Questions on Circular Motion

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