Physics Center Of Mass of a Two-Particle System

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Center Of Mass of a Two-Particle System

CENTRE OF MASS

The centre of mass of an object is defined as the point where whole mass of the object may be supposed to be concentrated and the point mass behaves similarly to an external force as the whole object would have behaved.

Characteristics of centre of mass

a. Position of centre of mass does not depend on Co-ordinate system chosen.

b. Position of centre of mass does not change due to internal forces.

c. External forces alone can change the position of centre of mass.

d. Centre of mass of a body may lie inside or outside the body.

e. Centre of mass always lie on the axis of symmetry of the body if it exists.

f. For a body in which there are two or more axes of symmetry, then centre of mass lies at their point of intersection.

g. In a system of N particles, the centre of mass of the system may or may not coincide with any of the particles.

h. Centre of mass is a point, about which the sum of moments of masses ( i.e. multiplication of mass and its position vector with respect to centre of mass) of all the particles in the body is zero.

If `m_1 , m_2 , m_3`.............. are situated at positions vector `vecr_1, vecr_2, vecr_3`................. with respect to the centre of mass, then `m_1vecr_1 , m_2vecr_2 , m_3vecr_3`..........................= 0

The centre of mass of an object is defined as the point where whole mass of the object may be supposed to be concentrated and the point mass behaves similarly to an external force as the whole object would have behaved.

Characteristics of centre of mass

a. Position of centre of mass does not depend on Co-ordinate system chosen.

b. Position of centre of mass does not change due to internal forces.

c. External forces alone can change the position of centre of mass.

d. Centre of mass of a body may lie inside or outside the body.

e. Centre of mass always lie on the axis of symmetry of the body if it exists.

f. For a body in which there are two or more axes of symmetry, then centre of mass lies at their point of intersection.

g. In a system of N particles, the centre of mass of the system may or may not coincide with any of the particles.

h. Centre of mass is a point, about which the sum of moments of masses ( i.e. multiplication of mass and its position vector with respect to centre of mass) of all the particles in the body is zero.

If `m_1 , m_2 , m_3`.............. are situated at positions vector `vecr_1, vecr_2, vecr_3`................. with respect to the centre of mass, then `m_1vecr_1 , m_2vecr_2 , m_3vecr_3`..........................= 0

Center of Mass of a Two-Particle System

Consider a system consisting of two point masses `m_1` and `m_2`, whose position vectors at a time t with reference to the origin `O` of the inertial frame are `vecr_1` and `vecr_2` respectively.

Then for a system of two particles their CM is defined as

`vecr_(cm) = (m_1vecr_1 + m_2vecr_2)/(m_1 + m_2)`

Shift in position of CM is

`trianglevecr_(cm) =(m_1triangle vecr_1 + m_2triangle vecr_2)/(m_1 + m_2)`

Where `triangle vecr_1` : shift in position of `m_1, trianglevecr_2`: shift in position of `m_2`

NOTE:

If `vecF_(ext) = 0` and initially CM is at rest, then for two particles system

`trianglevecr_(cm) = (m_1trianglevecr_1 + m_2trianglevecr_2)/(m_1 + m_2)`

`m_1trianglevecr_1 + m_2trianglevecr_2` = 0, i.e. CM will not shift.

Consider a system consisting of two point masses `m_1` and `m_2`, whose position vectors at a time t with reference to the origin `O` of the inertial frame are `vecr_1` and `vecr_2` respectively.

Then for a system of two particles their CM is defined as

`vecr_(cm) = (m_1vecr_1 + m_2vecr_2)/(m_1 + m_2)`

Shift in position of CM is

`trianglevecr_(cm) =(m_1triangle vecr_1 + m_2triangle vecr_2)/(m_1 + m_2)`

Where `triangle vecr_1` : shift in position of `m_1, trianglevecr_2`: shift in position of `m_2`

NOTE:

If `vecF_(ext) = 0` and initially CM is at rest, then for two particles system

`trianglevecr_(cm) = (m_1trianglevecr_1 + m_2trianglevecr_2)/(m_1 + m_2)`

`m_1trianglevecr_1 + m_2trianglevecr_2` = 0, i.e. CM will not shift.

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