A horizontal force F is applied at the centre of mass of a cylindrical object of mass m and radius R ,…

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?

  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.

Solution

Let moment of inertia =I

For pure rolling, a=αR

and F−f=ma       ...1

and fR=Iα

or fR=IaR

or f=IaR2

Therefore, F−IaR2=ma

∴   F=m+IR2a

⇒ For the same value of F, value of a depends on I.

⇒ For a solid cylinderI=mR22 (to calculate maximum a)

f=μmg

∴ μmg=mR22×aR2

Therefore, a=2μg

⇒f is dependent on applied force.

⇒ For thin-walled hollow cylinderI=mR2,

F=m+mR2R2a=2ma

∴  a=F2m 

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Asked in: JEE Advanced 2021 (Paper 1)

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