Question 580: Centre of Mass and Rotational Motion

Question

A molecule consists of two atoms each of mass '$$m$$' and separated by a distance '$$\mathrm{d}$$'. At room temperature, if the average rotational kinetic energy is '$$E$$' then the angular frequency is

Accepted Answer

To determine the angular frequency of a molecule consisting of two atoms, we need to use the concept of rotational kinetic energy. The average rotational kinetic energy of a diatomic molecule at room temperature is given by:
$$ E = \frac{1}{2} I \omega^2 $$
Here, $$E$$ is the average rotational kinetic energy, $$I$$ is the moment of inertia, and $$\omega$$ is the angular frequency.
The moment of inertia for the system can be calculated using the formula:
$$ I = \mu d^2 $$
where $$\mu$$ is the reduced mass of the two-atom system and $$d$$ is the separation distance. For two atoms of mass $$m$$, the reduced mass $$\mu$$ is given by:
$$ \mu = \frac{m \cdot m}{m + m} = \frac{m^2}{2m} = \frac{m}{2} $$
Thus, the moment of inertia becomes:
$$ I = \frac{m}{2} d^2 $$
Substituting $$I$$ back into the rotational kinetic energy equation, we get:
$$ E = \frac{1}{2} \left( \frac{m}{2} d^2 \right) \omega^2 $$
Simplifying this, we obtain:
$$ E = \frac{m d^2 \omega^2}{4} $$
To solve for the angular frequency $$\omega$$, we rearrange the equation:
$$ \omega^2 = \frac{4E}{m d^2} $$
Taking the square root of both sides, we find:
$$ \omega = \sqrt{\frac{4E}{m d^2}} $$
Which simplifies to:
$$ \omega = \frac{2}{d} \sqrt{\frac{E}{m}} $$
Thus, the correct answer is:
Option A:
$$\frac{2}{\mathrm{~d}} \sqrt{\frac{\mathrm{E}}{\mathrm{m}}}$$

Answer

$$\frac{2}{\mathrm{~d}} \sqrt{\frac{\mathrm{E}}{\mathrm{m}}}$$

Answer

$$\frac{\mathrm{d}}{2} \sqrt{\frac{\mathrm{m}}{\mathrm{E}}}$$

Answer

$$\sqrt{\frac{E d}{m}}$$

Answer

$$\sqrt{\frac{m}{E d}}$$

Centre of Mass and Rotational Motion

A molecule consists of two atoms each of mass '$m' and separated by a distance '\mathrm{d}'. At room temperature, if the average rotational kinetic energy is 'E$' then the angular frequency is

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