Multiple Select Questlon Select all that apply Two identical solid spheres $$ A $$ and $$ B $$ have the same mass and diameter. Both are rotating about lines passing through their centers with $$ K_{A}K_{B} $$. Select all of the following statements that are true. $$ \square \omega_{A}=\omega_{B} $$ $$ \square \omega_{A}\omega_{B} $$ $$ \square I_{B} $$ $$ \square I_{A}I_{B} $$ $$ \square $$ $$ I_{A}

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Multiple Select Questlon Select all that apply Two identical solid spheres $$ A $$ and $$ B $$ have the same mass and diameter. Both are rotating about lines passing through their centers with $$ K_{A}>K_{B} $$. Select all of the following statements that are true. $$ \square \omega_{A}=\omega_{B} $$ $$ \square \omega_{A}>\omega_{B} $$ $$ \square I_{B} $$ $$ \square I_{A}>I_{B} $$ $$ \square $$ $$ I_{A}

UpStudy AI · Accepted Answer

To analyze the problem, we need to understand the relationship between the rotational kinetic energy ($$ K $$), angular velocity ($$ \omega $$), and moment of inertia ($$ I $$) for the two identical solid spheres $$ A $$ and $$ B $$.

The rotational kinetic energy is given by the formula:
$$
K = \frac{1}{2} I \omega^2
$$

Since both spheres are identical, they have the same mass ($$ m $$) and diameter, which means they also have the same moment of inertia ($$ I $$). For a solid sphere, the moment of inertia is given by:
$$
I = \frac{2}{5} m r^2
$$
where $$ r $$ is the radius of the sphere.

Given that $$ K_A > K_B $$, we can analyze the implications:

1. **Moment of Inertia**: Since both spheres are identical, we have:
   $$
   I_A = I_B
   $$

2. **Rotational Kinetic Energy**: From the kinetic energy formula, we can express the kinetic energies of both spheres:
   $$
   K_A = \frac{1}{2} I_A \omega_A^2
   $$
   $$
   K_B = \frac{1}{2} I_B \omega_B^2
   $$
   Since $$ K_A > K_B $$ and $$ I_A = I_B $$, it follows that:
   $$
   \frac{1}{2} I_A \omega_A^2 > \frac{1}{2} I_B \omega_B^2
   $$
   This simplifies to:
   $$
   \omega_A^2 > \omega_B^2
   $$
   Therefore, we conclude that:
   $$
   \omega_A > \omega_B
   $$

Now, let's evaluate the statements:

1. $$ \square \omega_{A}=\omega_{B} $$ - **False** (since $$ \omega_A > \omega_B $$)
2. $$ \square \omega_{A}>\omega_{B} $$ - **True** (as derived above)
3. $$ \square I_{B} $$ - **Incomplete statement** (not enough information)
4. $$ \square I_{A}>I_{B} $$ - **False** (since $$ I_A = I_B $$)
5. $$ \square I_{A}\omega_{B} $$

The final selection is:
- $$ \square \omega_{A}>\omega_{B} $$ (True)
$$ \omega_{A} > \omega_{B} $$

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