If particles of masses `m_1, m_2, m_3`......... are moving with velocities `vecv_1, vecv_1, vecv_3`.........respectively, then velocity of centre of mass is given by
`vecV_(cm) = (m_1 vecv_1 + m_2 vecv_2 + ............)/(m_1 + m_2 + .................)`
= `(sum m_i vecv_i)/(sum m_i) = text(Total linear momentum of the system)/text(Total mass)`
Now components of velocity of centre of mass along X, Y, and Z axes can be written as.
`V_(cm_x) = (m_1v_(x_1) + m_2v_(x_2) + ............)/(m_1 + m_2 + .............)`
= `(sum m_i vecV_(x_i))/(sum m_i) = (sum P_(x_i))/(sum m_i)`
`V_(cm_y) = (m_1v_(y_1) + m_2v_(y_2) + .........)/(m_1 + m_2 + ..........) = (sum m_iV_(y_i))/(sum m_i) = (sum P_(y_i))/(sum m_i)`
`V_(cm_z) = (m_1v_(z_1) + m_2v_(z_2) +........)/(m_1 + m_2 +..........) = (sum m_iV_(z_i))/(sum m_i) = (sum P_(z_i))/(sum m_i)`
If `vecv_1 = vecv_2=........... = vecv` then `vecV_(cm) = vecv`
Note :
(a) Motion of centre of mass is not affected by the internal forces. Therefore, if a shell moving under gravity explodes into pieces moving in different direction, even then the centre of mass moves along the same (previous )path.
(b) Due to mutual interaction forces, velocity of CM does not change.
(c) If no external force acts on the body, state of motion of its centre of mass remains the same i.e. if it is moving with some velocity then it keep on moving in the same direction with same speed and if CM is at rest, it will remain at rest.
If particles of masses `m_1, m_2, m_3`......... are moving with velocities `vecv_1, vecv_1, vecv_3`.........respectively, then velocity of centre of mass is given by
`vecV_(cm) = (m_1 vecv_1 + m_2 vecv_2 + ............)/(m_1 + m_2 + .................)`
= `(sum m_i vecv_i)/(sum m_i) = text(Total linear momentum of the system)/text(Total mass)`
Now components of velocity of centre of mass along X, Y, and Z axes can be written as.
`V_(cm_x) = (m_1v_(x_1) + m_2v_(x_2) + ............)/(m_1 + m_2 + .............)`
= `(sum m_i vecV_(x_i))/(sum m_i) = (sum P_(x_i))/(sum m_i)`
`V_(cm_y) = (m_1v_(y_1) + m_2v_(y_2) + .........)/(m_1 + m_2 + ..........) = (sum m_iV_(y_i))/(sum m_i) = (sum P_(y_i))/(sum m_i)`
`V_(cm_z) = (m_1v_(z_1) + m_2v_(z_2) +........)/(m_1 + m_2 +..........) = (sum m_iV_(z_i))/(sum m_i) = (sum P_(z_i))/(sum m_i)`
If `vecv_1 = vecv_2=........... = vecv` then `vecV_(cm) = vecv`
Note :
(a) Motion of centre of mass is not affected by the internal forces. Therefore, if a shell moving under gravity explodes into pieces moving in different direction, even then the centre of mass moves along the same (previous )path.
(b) Due to mutual interaction forces, velocity of CM does not change.
(c) If no external force acts on the body, state of motion of its centre of mass remains the same i.e. if it is moving with some velocity then it keep on moving in the same direction with same speed and if CM is at rest, it will remain at rest.