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System of Particles and Rotational Motion

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Physics Part I

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12
A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. Find the angular speed of the door just after the bullet embeds into it.

(Hint: The moment of inertia of the door about the vertical axis at one end is ML2/3.)


Mass of the bullet, 
m = 10 g = 10 × 10–3 kg 

Velocity of the bullet, v = 500 m/s 

Thickness of the door, L = 1 m 

Radius of the door, r = m / 2 

Mass of the door, M = 12 kg 

Angular momentum imparted by the bullet on the door, 

    α = mvr 

      = (10 × 10-3 ) × (500) × 1 half

      =  2.5 kg m
2 s-1                                   ...(i) 

Moment of inertia of the door is given by, 

I thin space equals space ML squared over 3

space space equals space open parentheses 1 third close parentheses space x space 12 space x space 1 squared space

space space equals space 4 space k g space m squared space

B u t comma space

alpha space equals space I space omega space

omega equals space alpha over I
space space space equals space fraction numerator 2.5 over denominator 4 end fraction space
space space space equals space 0.625 space r a d divided by s e c  

This is the required angular speed of the door just after the bullet embeds into it. 


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. 

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.

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. 

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.