Question

Two points masses of mass $4m$ and $m$ respectively separated by distance r, are revolved under mutual force of attraction. Ratios of their kinetic energies will be-
A. 1:4
B. 1:5
C. 1:1
D. 1:2

Answer 1

Hint: The masses of two points are 4m and m. The distance between them is given r. First, we will find the position of the centre of the mass by using the formula $x = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}}$, where x is the position.
Formula used: $x = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}}$, $K = \dfrac{1}{2}I{\omega ^2}$, $I = m{r^2}$

Complete Step-by-Step solution:
Considering the origin of the coordinate system at 4m, we evaluate the position of the centre of mass as-
Value of ${m_1} = 4m$
Value of ${m_2} = m$
Value of ${x_1} = 0$
Value of ${x_2} = r$
Applying the formula $x = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}}$, we get-
$
   \Rightarrow \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}} \\
    \\
   \Rightarrow \dfrac{{4m \times 0 + m \times r}}{{4m + m}} \\
    \\
   \Rightarrow \dfrac{{mr}}{{5m}} \\
$
$ \Rightarrow \dfrac{r}{5}$ (equation 1)
Thus, the centre of mass is $\dfrac{r}{5}$​ from 4m. So, the distance from mass m will be-
 $ \Rightarrow r - \dfrac{r}{5}$
$ \Rightarrow \dfrac{{4r}}{5}$ (equation 2)
The formula of kinetic energy is calculated as $K = \dfrac{1}{2}I{\omega ^2}$.
The angular velocity of both the particles will be the same as they are revolving around mutual force of attention.
The ratio of the kinetic energy of both the particles will be-
$
   \Rightarrow \dfrac{{{K_{4m}}}}{{{K_m}}} = \dfrac{{\dfrac{1}{2}{I_{4m}}{\omega ^2}}}{{\dfrac{1}{2}{I_m}{\omega ^2}}} \\
    \\
   \Rightarrow \dfrac{{{K_{4m}}}}{{{K_m}}} = \dfrac{{{I_{4m}}}}{{{I_m}}} \\
$
As we know, the formula for $I = m{r^2}$. Putting the values of equation 1 and equation 2 accordingly, we get-
So,
$
   \Rightarrow \dfrac{{{K_{4m}}}}{{{K_m}}} = \dfrac{{{I_{4m}}}}{{{I_m}}} = \dfrac{{4m{{\left( {\dfrac{r}{5}} \right)}^2}}}{{m{{\left( {\dfrac{{4r}}{5}} \right)}^2}}} \\
    \\
   \Rightarrow \dfrac{{{K_{4m}}}}{{{K_m}}} = \dfrac{1}{4} \\
$
Hence, option A is the correct option.

Note: Rotational energy occurs because of the object's rotation and is a part of its absolute kinetic energy. On the off chance that the rotational energy is considered separately across an object's axis of rotation, the moment of inertia is observed. Rotational energy also known as angular kinetic energy which is characterized as- The kinetic energy because of the rotation of an object and is a piece of its complete kinetic energy.