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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
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B
gsinθ1+MR2/I
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C
gsinθ1I/MR2
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D
gsinθ1MR2/I
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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), $$mg\sin\theta-\dfrac{Ia}{R^2}= ma$$
$$\implies a= \dfrac{g\sin\theta}{1+\dfrac{I}{MR^2}}$$

438119_40791_ans_35790b6e5def4788bcb8bae436102123.png

Physics

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