A horizontal force F is applied at the centre of mass of a cylindrical object...

A horizontal force $F$ is applied at the centre of mass of a cylindrical object of mass $m$ and radius $R$ , perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $μ$ . The centre of mass of the object has an acceleration $a$ . The acceleration due to gravity is $g$ . Given that the object rolls without slipping, which of the following statement(s) is (are) correct?

For the same

F

, the value of

a

does not depend on whether the cylinder is solid or hollow.

For a solid cylinder, the maximum possible value of

a

2 μ g

The magnitude of the frictional force on the object due to the ground is always

μ m g

For a thin-walled hollow cylinder,

a = \frac{F}{2 m}

Solution:

Let moment of inertia $= I$

For pure rolling, $a = α R$

and $F - f = m a . . . (1)$

and $f R = I α$

or $f R = \frac{I a}{R}$

or $f = \frac{I a}{R^{2}}$

Therefore, $F - \frac{I a}{R^{2}} = m a$

$∴ F = (m + \frac{I}{R^{2}}) a$

$\Rightarrow$ For the same value of $F$ , value of $a$ depends on $I$ .

$\Rightarrow$ For a solid cylinder $(I = \frac{m R^{2}}{2})$ (to calculate maximum $a$ )

$f = μ m g$

$∴ μ m g = (\frac{m R^{2}}{2}) \times \frac{a}{R^{2}}$

Therefore, $a = 2 μ g$

$\Rightarrow f$ is dependent on applied force.

$\Rightarrow$ For thin-walled hollow cylinder $(I = m R^{2})$ ,

$F = (m + \frac{m R^{2}}{R^{2}}) a = 2 m a$

$∴ a = \frac{F}{2 m}$

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A horizontal force F is applied at the centre of mass of a cylindrical object...

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