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Centripetal Acceleration

Two balls of ...

Question

Two balls of masses $1 k g$ each are attached with two strings each of length $1 m$ as shown in figure. They are rotated in a horizontal circle with angular speed $ω = 4 r a d / s$ about point 'O'. The ratio of tensions $\frac{T_{1}}{T_{2}}$ is:

\frac{3}{2}

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\frac{2}{3}

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\frac{1}{4}

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Solution

The correct option is A $\frac{3}{2}$
Given $m_{A} = 1 k g; m_{B} = 1 k g$
$r_{A} = length of m_{A} from 'O' = 1 m$
$r_{B} = length of m_{B} {from 'O'}^{'} = 2 m$

FBD considering only horizontal forces, because vertical forces balance each other i.e $N = m g$ :

Applying equation of circular motion along radial direction for $m_{B}$ and $m_{A}$ respectively:
$T_{2} = m_{B} r_{B} ω^{2} = 1 \times 2 \times 4^{2} = 32 N - (1)$
and
$T_{1} - T_{2} = m_{A} r_{A} ω^{2} = 1 \times 1 \times (4)^{2} = 16 N - (2)$
Adding (1) and (2),
$T_{1} = 48 N$
$∴ \frac{T_{1}}{T_{2}} = \frac{48}{32} = \frac{3}{2}$

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