Question

# A round uniform body of radius $$R$$, mass $$M$$ and moment of inertia $$I$$, rolls down (without slipping) an inclined plane making an angle with the horizontal. Then its acceleration is :

A
gsinθ1+I/MR2
B
gsinθ1+MR2/I
C
gsinθ1I/MR2
D
gsinθ1MR2/I

Solution

## The correct option is A $$\dfrac{g\sin\theta }{1+I/MR^{2}}$$For translational motion:  $$F=Ma$$$$mgsin\theta-f=ma$$            (1)For rotational motion: Net torque about O, $$\tau= I\alpha$$  $$f R= I\alpha$$$$\implies f=\dfrac{I\alpha}{R}$$Also body rolls without slipping, thus  $$a= r\alpha$$Hence, $$f=\dfrac{Ia}{R^2}$$Putting it in (1), $$mg\sin\theta-\dfrac{Ia}{R^2}= ma$$$$\implies a= \dfrac{g\sin\theta}{1+\dfrac{I}{MR^2}}$$Physics

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