6.System of Particles and Rotational Motion
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એક ઘન ગોળો ગબડતી ગતિમાં છે.ગબડતિ ગતિ (લોટણ ગતિ) માં પદાર્થ સ્થાનાંતરીત ગતિઊર્જા $(K_t) $ અને ભ્રમણીય ગતિઊર્જા $(K_r)$ એક સાથે ધરાવે છે.આ ગોળા માટે $ K_t: (K_t+ K_r)$ નો ગુણોત્તર છે.

A

$7:10$

B

$5:7$

C

$2:5$

D

$10:7$

(NEET-2018) (AIPMT-1991)

Solution

       Translational kinetic energy, ${K_t} = \frac{1}{2}m{v^2}$

Rotational kinetic energy, ${K_r} = \frac{1}{2}I{\omega ^2}$

$\begin{array}{ccccc}
\therefore \,{K_t} + {K_r} = \frac{1}{2}m{v^2} + \frac{1}{2}I{\omega ^2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2}m{v^2} + \frac{1}{2}\left( {\frac{2}{5}m{r^2}} \right){\left( {\frac{v}{r}} \right)^2}\\
\therefore \,{K_t} + {k_r} = \frac{7}{{10}}m{v^2}\,\,\,\,\,\,\left[ {I = \frac{2}{5}m{r^2}\left( {for\,sphere} \right)} \right]\\
So,\,\frac{{{K_t}}}{{{K_t} + {K_r}}} = \frac{5}{7}
\end{array}$

Standard 11
Physics

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