A horizontal force is applied at the center of mass of a cylindrical object of mass and radius , perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is . The center of mass of the object has an

Question

A horizontal force $F$ is applied at the center 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 center 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 [2021]

Smart Achievers · Accepted Answer

(2, 4)

For solid cylinder,

$F \times R = \frac{3}{2} m R^{3} \times (\frac{a}{R}) \Rightarrow a = \frac{2 F}{3 m}$

For hollow cylinder,

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

For solid cylinder

$f = F - m \times \frac{2 F}{3 m} = \frac{F}{3} \leq μ m g \Rightarrow F \leq 3 μ m g$

$∴$ $a_{\max} = 2 μ g$

Solution

1 For the same $F$ , the value of $a$ does not depend on whether the cylinder is solid or hollow

2 For a solid cylinder, the maximum possible value of $a$ is $2 μ g$

3 The magnitude of the frictional force on the object due to the ground is always $μ m g$

4 For a thin-walled hollow cylinder, $a = \frac{F}{2 m}$

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Solution

1 For the same F, the value of a does not depend on whether the cylinder is solid or hollow

2 For a solid cylinder, the maximum possible value of a is 2μg

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

4 For a thin-walled hollow cylinder, a=F2m

Ans. (2, 4) For solid cylinder, F×R=32mR3×(aR) ⇒ a=2F3m For hollow cylinder, F×R=(2mR2)×aR ⇒ a=F2m For solid cylinder f=F-m×2F3m=F3≤μmg ⇒ F≤3μmg ∴ amax=2μg

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1 For the same $F$ , the value of $a$ does not depend on whether the cylinder is solid or hollow

2 For a solid cylinder, the maximum possible value of $a$ is $2 μ g$

3 The magnitude of the frictional force on the object due to the ground is always $μ m g$

4 For a thin-walled hollow cylinder, $a = \frac{F}{2 m}$