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in linear momentum, considering the collision of 2 particles to deudce that the momentum in a closed system is concerved

(W_1 + F_12)t = m.v_f1 - m.v_01 ....(i)

where

W_1=external force during impact

F_12=force acting on body 1 by body 2 in collision

t=impacting time

m.v_f1= momentum of body 1 after collision

m.v_01=momentum of body 1 before collision

similiarly

(W_2 + F_21)t = m.v_f2 - m.v_02 ....(ii)

adding 1 and 2

since F_12 = -F_21 by newton's third law

therefore (W_1+W_2)t = (m.v_f1 +m.v_f2) - (m.v_01 + m.v_02)

if the particles is isolated system, W_1+W_2=0 and the linear momentum of the system is conserved

similary result can be deueced by adding more of these equations of same form of (i) and (ii)

with calculus

F=dp/dt=d(mv)/dt= m. dv/dt= ma

with no external force a=o and linear momentum is conseved

I get how this works similiarly with rotational dynamics only in COLLISION

in the first method, conservation of angular momentum closed system durning collsion can be dudeuced by the 1st method by replacing

W_1 by torque by exteral force

F_12 by T_12= torque created on body(or disc) 1 by body 2

m.v_f1 by I.w_f1= angular momentum of body after collision

m.v_01 by I.w_01= angular momentum of body before collision

and by calculus

T= dL/dt = d(Iw)/dt =I. dw/dt = Ia

this is what i dont get: in the case of a ice skater, pulling arms in during rotation, I(moment of inertia) is changed, so how is conservation of momentum explained with calculus method when I of the system changes and i dont get at all how the non calculus method explains the conservation in this case (i only get it if it is during impact in rotational but in linear if totally understand how this method explains the conservation whether it impact of no impact)

(W_1 + F_12)t = m.v_f1 - m.v_01 ....(i)

where

W_1=external force during impact

F_12=force acting on body 1 by body 2 in collision

t=impacting time

m.v_f1= momentum of body 1 after collision

m.v_01=momentum of body 1 before collision

similiarly

(W_2 + F_21)t = m.v_f2 - m.v_02 ....(ii)

adding 1 and 2

since F_12 = -F_21 by newton's third law

therefore (W_1+W_2)t = (m.v_f1 +m.v_f2) - (m.v_01 + m.v_02)

if the particles is isolated system, W_1+W_2=0 and the linear momentum of the system is conserved

similary result can be deueced by adding more of these equations of same form of (i) and (ii)

with calculus

F=dp/dt=d(mv)/dt= m. dv/dt= ma

with no external force a=o and linear momentum is conseved

I get how this works similiarly with rotational dynamics only in COLLISION

in the first method, conservation of angular momentum closed system durning collsion can be dudeuced by the 1st method by replacing

W_1 by torque by exteral force

F_12 by T_12= torque created on body(or disc) 1 by body 2

m.v_f1 by I.w_f1= angular momentum of body after collision

m.v_01 by I.w_01= angular momentum of body before collision

and by calculus

T= dL/dt = d(Iw)/dt =I. dw/dt = Ia

this is what i dont get: in the case of a ice skater, pulling arms in during rotation, I(moment of inertia) is changed, so how is conservation of momentum explained with calculus method when I of the system changes and i dont get at all how the non calculus method explains the conservation in this case (i only get it if it is during impact in rotational but in linear if totally understand how this method explains the conservation whether it impact of no impact)

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