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System of Particles and Rotational Motion

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Physics Part I

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Class 10 Class 12
A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.

Mass of the hollow cylinder, m = 3 kg 

Radius of the hollow cylinder, r = 40 cm = 0.4 m

Applied force, F = 30 N 

The moment of inertia of the hollow cylinder about its geometric axis, 

I = mr

  = 3 × (0.4)2 

  = 0.48 kg m

Torque, τ = F × r 

             =  30 × 0.4 

             = 12 Nm 

For angular acceleration α, torque is also given by the relation, 

   τ = Iα 

   α = τ / I 

     =  12 / 0.48

     = 25 rad s-2 

Linear acceleration = τα 

                         = 0.4 × 25

                         = 10 m s–2 

This is the required linear acceleration of the rope. 

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. 

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. 

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. 

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. 

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.