﻿ A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30°. The coefficient of static friction µs = 0.25. (a) How much is the force of friction acting on the cylinder? (b) What is the work done against friction during rolling? (c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly? from Physics System of Particles and Rotational Motion Class 11 Manipur Board

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A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30°. The coefficient of static friction µs = 0.25.

(a) How much is the force of friction acting on the cylinder?

(b) What is the work done against friction during rolling?

(c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly?

Mass of the cylinder, m = 10 kg

Radius of the cylinder, r = 15 cm = 0.15 m

Co-efficient of kinetic friction, µ= 0.25

Angle of inclination, θ = 30°

Moment of inertia of a solid cylinder about its geometric axis, I = mr2
The various forces acting on the cylinder are shown in the following figure,

The acceleration of the cylinder is given as,

a = mg Sinθ / [m + (I/r2) ]

= mg Sinθ / [m + {(1/2)mr2/ r2} ]

=  g Sin 30°

= × 9.8 × 0.5

=  3.27 ms-2

(a) Using Newton’s second law of motion,

Net force, fnet = ma

mg Sin 30° - f = ma

f = mg Sin 30° - ma

= 10 × 9.8 × 0.5 - 10 × 3.27

49 - 32.7 = 16.3 N

(b) During rolling, the instantaneous point of contact with the plane comes to rest.

Hence, the work done against frictional force is zero.

(c) For rolling without skid,

Using the relation,

μ = (1/3) tan θ

tan θ = 3μ

= 3 × 0.25

∴ θ = tan-1 (0.75)

= 36.87°.

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