"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?" Question 7.31System Of Particles And Rotational Motion.
Answers
Answer:
Mass of the cylinder, m = 10 kg
Radius of the cylinder, r = 15 cm = 0.15 m
Co-efficient of kinetic friction, µk = 0.25
Angle of inclination, θ = 30°
Moment of inertia of a solid cylinder about its geometric axis, I = (1/2)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} ]
= (2/3) g Sin 30°
= (2/3) × 9.8 × 0.5 = 3.27 ms-2
(a) Using Newton’s second law of motion, we can write net force as:
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, we have the relation:
μ = (1/3) tan θ
tan θ = 3μ = 3 × 0.25
∴ θ = tan-1 (0.75) = 36.87°.
- mass of cylinder (m) = 10 Kg
- Radius (r) = 15 cm = 0.15 m
- Inclination of plane (∅) = 30°
- coefficient of friction (u) = 0.25
- friction = mgsin∅/(1 + mr²/I)
- For solid cylinder (I) = mr²/2
- Fr = mgsin∅/3
- = 10 × 9.8 × 1/2 × 1/3
- = 16.3 N
- force of friction acts perpendicular to the direction of the displacement.
- So , work done against during rolling .
- W = Fscos90° = 0
- for rolling without slipping use formula,
- for rolling without slipping use formula,u = tan∅/( 1 + mr²/I)
- For solid cylinder
- u = tan∅/3
- tan∅ = 3u
- = 3×0.25 = 0.75 = tan36°54'
- ∅ = 36°54' ≈ 37°