Velocity of centre of mass

Velocity of centre of mass

Velocity of center of mass

Velocity of centre of mass :  Velocity of centre of mass of any system is the velocity of the point where all the mass of the system is concentrated. Let's understand this concept with the help of diagram given below.

From the above fig, Consider a system rotating about O and  made up of different particle having mass m_1,m₂,  m_n, at a distance of  r₁ , r₂, r_n   from Origin  ' O '  and having velocity v₁ , v₂,  v_n .Let M be the centre of mass of the system situated at a distance R from the point Oand having velocity V.

The linear momentum for each particle is given by equation ,

m₁v₁ ; m₂v₂ ; m₃v₃ ; m_nv_n

The total moment of  force due to each particle is given by equation ,

m₁v₁  + m₂v₂ + m₃v₃  +m_nv_n------(1)

The  linear momentum due to centre of mass is given by equation ,

MV ------(2)

From (1) and (2)

MV = _{m1v1  + m2v2 + m3v3  +mnvn}

V = [(m₁v₁  + m₂v₂ + m₃v₃  +m_nv_{n) / M]---(A)}

_{Equation can be written as}

_{Equation A gives the velocity of centre of mass of a system .}

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_{Acceleration of centre of mass}

acceleration of Centre of mass

Acceleration of centre of mass :  acceleration of centre of mass of any system is the acceleration of the point where all the mass of the system is concentrated. Let's understand this concept with the help of diagram given below.

From the above fig, Consider a system rotating about O and  made up of different particle having mass m_1,m₂,  m_n, at a distance of  r₁ , r₂, r_n   from Origin  ' O '  and having linear acceleration a₁ , a₂,  a_n .Let M be the center of mass of the system situated at a distance R from the point O and having acceleration 'a'.

The force for each particle is given by equation ,

m₁a₁ ; m₂a₂ ; m₃a₃ ; m_na_n

The total  force due to each particle is given by equation ,

m₁a₁  + m₂a₂ + m₃a₃  +m_na_n------(1)

The force on centre of mass is given by equation ,

Ma ------(2)

From (1) and (2)

Ma = m₁a₁  + m₂a₂ + m₃a₃  +m_na_n

a = [(m₁a₁  + m₂a₂ + m₃a₃  +m_na_{n) / M]---(A)}

│<<<Torque│    _{│Center of gravity>>>│}

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