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System of Particles and Rotational Motion

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Physics Part I

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Physics

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Class 10 Class 12
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 =open parentheses 1 half close parentheses 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} ] 

      = open parentheses 2 over 3 close parentheses g Sin 30° 

      =open parentheses 2 over 3 close parentheses × 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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What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system. 
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What is the need of centre of mass?

Newton’s second law of motion is strictly applicable to point masses only. To apply the Newton's law of motion to rigid bodies, the concept of centre of mass is introduced.

The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object. 
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Is it necessary for centre of mass to lie within the body?

No, centre of mass needs not to lie within the body. It is not necessary that the total mass of the system be actually present at the centre.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion. 
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Is it necessary that there should be matter at the centre of mass of system?

No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.
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Define centre of mass.

Centre of mass of a body or a system of bodies is a point at which the entire mass of the body or system is supposed to be concentrated. 
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