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A cylinder of mass M_c and sphere of mass M_s are placed at points A and B of two inclines, respectively.

(See Figure). If they roll on the incline without slipping such that their accelerations are the same, then

The ratio $\frac{\sin \Theta _{c}}{\sin \Theta _{s}}$ is

Option 1)
$\sqrt{\frac{8}{7}}$

Option 2)
$\sqrt{\frac{15}{14}}$

Option 3)
$\frac{8}{7}$

Option 4)
$\frac{15}{14}$

Answers (1)

best_answer

As we have learned

Rolling of a body on an inclined plane -

$a= \frac{g\sin \Theta }{1+\frac{K^{2}}{R^{2}}}$

$f= \frac{mg\sin \Theta }{1+\frac{R^{2}}{K^{2}}}$

- wherein

K=Radius of gyration

$\Theta$ = Angle of inclination

Acceleration along inclined plane

$a = \frac{g \sin \theta }{1+ \frac{Kc^2m}{R^2}}$

FOr sphere

$Kc^2m = 2/5 R^2 \Rightarrow a_l = \frac{g \sin \theta }{1+ 2/5 }$

$a_s = 5 /4 g \sin \theta$

For cylinder

$Kc^2m = 1/2 R^2 \Rightarrow a_c = \frac{g \sin \theta }{1+1/2}$

$a_c = 2/3 g \sin \theta _c$

$a_s = a_c \Rightarrow 5/7 (g \sin \theta _s) = 2/3 (g \sin \theta _c )$

$\frac{\sin \theta _c}{\sin \theta _s}= 15/14$

Option 1)

$\sqrt{\frac{8}{7}}$

Option 2)

$\sqrt{\frac{15}{14}}$

Option 3)

$\frac{8}{7}$

Option 4)

$\frac{15}{14}$

Posted by

Avinash

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