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# Acceleration ($a$) of a body of radius R rolling down an inclined plane of inclination is given by?

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## Step 1: Given data$K$ = Radius of gyration$R$ = Radius of the object$I$ = Inertia of the object$N$ = Normal force acting on the body$f$ = Frictional forceStep 2: Diagram From the figure, $mg\mathrm{sin}\theta &mg\mathrm{cos}\theta$=components of force due to body's weight ($mg$)Where, $m$=mass of the object, $g$= acceleration due to gravity, and $\theta$ = angle of the inclined planeStep 3: Calculating accelerationFrom the figure, we can see that the sine component of the weight of the object, $mg\mathrm{sin}\theta$ acts towards left and the frictional force $f$ acts opposite to the direction of motion. Since the body is rolling down, the resultant force, $ma=mg\mathrm{sin}\theta -f------\left(1\right)$ Here, the body is rolling due to friction, therefore net torque is given by,Torque = force $×$perpendicular distance,$\tau =fR$Also, Torque = moment of inertia $×$angular acceleration$\tau =I\alpha$$\therefore fR=mK²×\alpha \left(\because I=mK²\right)-------\left(2\right)$$K$ = radius of gyration$R$ = radius of the object$I$ = Inertia of the object$N$ = Normal force acting on the body$f$ = frictional forceIn relation to linear and angular quantities, since there is no slipping$v=R\omega$$a=R\alpha$----(3)$\alpha$=angular acceleration of the object$v$=linear velocity of the object$\omega$=angular velocity of the objectSolving 1,2 &3 simultaneously, From 2 Value of $f=\frac{m{K}^{2}\alpha }{R}$substitute this value in 1$mg\mathrm{sin}\left(\theta \right)-\frac{m{k}^{2}\alpha }{R}=ma\phantom{\rule{0ex}{0ex}}g\mathrm{sin}\left(\theta \right)-\frac{{K}^{2}\alpha }{R}=a$-----(4)From 3 $\alpha =\frac{a}{R}$substitute in 4We get, $a=\frac{g\mathrm{sin}\theta }{1+\frac{{K}^{2}}{{R}^{2}}}$Hence, the acceleration of a body rolling down an inclined plane is :$\mathbit{a}\mathbf{=}\frac{\mathbf{g}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }}{\mathbf{\left[}\mathbf{1}\mathbf{+}\frac{{K}^{2}}{{R}^{2}}\mathbf{\right]}}$  Suggest Corrections  0      Similar questions  Explore more