Question

An inclined plane makes an angle of∘30∘ with the horizontal. A solid cylinder rolling down this inclined plane from rest without slipping has a linear acceleration equal to:​

Answer 1

Given :

A solid cylinder is rolling down on inclined plane from rest without slipping.

Angle of inclination = 30°

To Find :

Linear acceleration of the cylinder.

Solution :

Moment of inertia of a solid cylinder about an axis passing through its centre and perpendicular to base is given by, I = MR²/2

$\sf:\implies\:Mk^2=MR^2/2$

• k = Radius of curvature

$\sf:\implies\:k^2=R^2/2$

$\bf:\implies\:k^2/R^2=1/2$

Applying 3rd equation of kinematics;

$\sf:\implies\:v^2-u^2=2ad$

$\sf:\implies\:\left(\sqrt{\dfrac{2gh}{1+\dfrac{k^2}{R^2}}}\right)^2-0^2=2ad$

$\sf:\implies\:\dfrac{2gh}{1+\dfrac{k^2}{R^2}}=2a\times\dfrac{h}{sin\theta}$

$:\implies\:\underline{\boxed{\bf{\orange{a=\dfrac{g\:sin\theta}{1+\dfrac{k^2}{R^2}}}}}}$

By substituting the given values;

$\sf:\implies\:a=\dfrac{10\times sin30^{\circ}}{1+\dfrac{1}{2}}$

$\sf:\implies\:a=\dfrac{10/2}{3/3}$

$:\implies\:\underline{\boxed{\bf{\purple{a=3.33\:ms^{-2}}}}}$

Answer 2

Given :

A solid cylinder is rolling down on inclined plane from rest without slipping.

Angle of inclination = 30°

To Find :

Linear acceleration of the cylinder.

Solution :

Moment of inertia of a solid cylinder about an axis passing through its centre and perpendicular to base is given by, I = MR²/2

$\sf:\implies\:Mk^2=MR^2/2$

k = Radius of curvature

$\sf:\implies\:k^2=R^2/2$

$\bf:\implies\:k^2/R^2=1/2$

Applying 3rd equation of kinematics;

$\sf:\implies\:v^2-u^2=2ad$

$\sf:\implies\:\left(\sqrt{\dfrac{2gh}{1+\dfrac{k^2}{R^2}}}\right)^2-0^2=2ad$

$\sf:\implies\:\dfrac{2gh}{1+\dfrac{k^2}{R^2}}=2a\times\dfrac{h}{sin\theta}$

$:\implies\:\underline{\boxed{\bf{\orange{a=\dfrac{g\:sin\theta}{1+\dfrac{k^2}{R^2}}}}}}$

By substituting the given values;

$\sf:\implies\:a=\dfrac{10\times sin30^{\circ}}{1+\dfrac{1}{2}}$

$\sf:\implies\:a=\dfrac{10/2}{3/3}$

$:\implies\:\underline{\boxed{\bf{\purple{a=3.33\:ms^{-2}}}}}$