Equal torques are applied about a central axis on two rings of same mass and same thickness but made up of different materials. If ratio of their densities is $4 : 1$ then the ratio of their angular acceleration will be :

4 : 1

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1 : 16

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8 : 1

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1 : 12

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Solution

The correct option is A $4 : 1$
The ring have same mass. Thus
$ρ_{1} V_{1} = ρ_{2} V_{2}$
or
$ρ_{1} t_{1} A_{1} = ρ_{2} t_{2} A_{2}$
or
$ρ_{1} t_{1} π r_{1}^{2} = ρ_{2} t_{2} π r_{2}^{2}$
As thickness is same we get
$\frac{ρ_{1}}{ρ_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}} = \frac{4}{1}$
Now as torque is same we get
$I_{1} α_{1} = I_{2} α_{2}$
or
$\frac{r_{2}^{2}}{r_{1}^{2}} = \frac{α_{1}}{α_{2}} = \frac{4}{1}$

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