In the given figure, two wheels $$P$$ and $$Q$$ are connected by a belt $$B$$. The radius of $$P$$ is three times that of $$Q$$. In the case of the same rotational kinetic energy, the ratio of rotational inertias $$\left(\frac{I_1}{I_2}\right)$$ will be $$x : 1$$. The value of $$x$$ will be _________.

We need to find the value of $$x$$ representing the ratio of the rotational inertias ($$\frac{I_1}{I_2}$$) for two connected wheels having the same rotational kinetic energy.

1. Analyze the Belt Connection (Linear Velocity)

From the problem, the two wheels $$P$$ and $$Q$$ are linked by a non-slipping belt $$B$$. Because the belt moves at a uniform speed, the linear tangential velocities at the rims of both wheels must be equal:

$$v_P = v_Q$$

Using the rotational relationship $$v = r\omega$$, we can rewrite this equality as:

$$r_1\omega_1 = r_2\omega_2$$

Where:

• $$r_1 = 3R$$ (radius of wheel $$P$$)
• $$r_2 = R$$ (radius of wheel $$Q$$)
• $$\omega_1$$ and $$\omega_2$$ are their respective angular velocities.

Substituting the radius values gives:

$$(3R)\omega_1 = R\omega_2 \implies \frac{\omega_1}{\omega_2} = \frac{1}{3}$$

2. Apply the Rotational Kinetic Energy Condition

The problem states that both wheels possess the same rotational kinetic energy ($$K_{\text{rot}}$$). The formula for rotational kinetic energy is:

$$K_{\text{rot}} = \frac{1}{2}I\omega^2$$

Equating the kinetic energies of both wheels:

$$\frac{1}{2}I_1\omega_1^2 = \frac{1}{2}I_2\omega_2^2$$

3. Calculate the Ratio of Rotational Inertias ($$x$$)

Rearranging the kinetic energy equation to isolate the ratio of moments of inertia ($$\frac{I_1}{I_2}$$):

$$\frac{I_1}{I_2} = \left(\frac{\omega_2}{\omega_1}\right)^2$$

Since we found earlier that $$\frac{\omega_1}{\omega_2} = \frac{1}{3}$$, its reciprocal is:

$$\frac{\omega_2}{\omega_1} = 3$$

Now, substitute this value back into our ratio equation:

$$\frac{I_1}{I_2} = (3)^2 = 9$$

The problem states that the ratio $$\frac{I_1}{I_2}$$ is expressed as $$x : 1$$. Comparing the terms:

$$x = 9$$

Conclusion

The value of $$x$$ is 9.