Two bodies rotate with kinetic energies E 1 and E 2 , moment of inertia about their axes of rotation is I 1 and I 2 . If 3 I 1 = I 2 and E 1 = 27 E 2 then the ratio of the angular mome

Question

Two bodies rotate with kinetic energies E 1 ​ and E 2 ​ , moment of inertia about their axes of rotation is I 1 ​ and I 2 ​ . If 3 I 1 ​ = I 2 ​ and E 1 ​ = 27 E 2 ​ then the ratio of the angular momenta L 1 ​ to L 2 ​ is

TestHub · Accepted Answer

The rotational kinetic energy is given by $E = \frac{L ^{2}}{2 I}$ , where $L$ is the angular momentum and $I$ is the moment of inertia.

Rearranging for $L$ , we get $L = 2 I E$ .

The ratio of the angular momenta is:

$\frac{L _{1}}{L _{2}} = \frac{I _{1} E _{1}}{I _{2} E _{2}}$

Given $3 I_{1} = I_{2} \Rightarrow \frac{I _{1}}{I _{2}} = \frac{1}{3}$ and $E_{1} = 27 E_{2} \Rightarrow \frac{E _{1}}{E _{2}} = 27$ .

Substituting these values:

$\frac{L _{1}}{L _{2}} = \frac{1}{3} \times 27 = 9 = 3$

Thus, the ratio $L_{1} : L_{2}$ is $3 : 1$ .

Answer: $3 : 1$

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