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Pipeng : Physics Translation Motion : Center Of Mass BETA Physics Homework Module

Center Of Mass BETA Physics Module

Description : Physics translation motion homework exercises : center of mass

Tools In This Module:

BETA : Translation : Center of Mass 01 : Center Of Mass : Linear Position And Velocity Of Center Of Mass : BETA Physics Homework Exercise
BETA : Translation : Center of Mass 02 : Center Of Mass : Linear Relative Velocity Of Center Of Mass : BETA Physics Homework Exercise
BETA : Translation : Center of Mass 03 : Center Of Mass : Linear Collision Masses Separate After Collision : BETA Physics Homework Exercise
BETA : Translation : Center of Mass 11 : Center Of Mass : Vector Position Of Center of Mass : BETA Physics Homework Exercise
BETA : Translation : Center of Mass 12 : Center Of Mass : Vector Velocity Of Center of Mass : BETA Physics Homework Exercise


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Module List

BETA : Translation : Center of Mass 01 : Center Of Mass : Linear Position And Velocity Of Center Of Mass : BETA Physics Homework Exercise

Description : Position and velocity of the center of mass for a linear two mass system.

Discussion : The center of mass is closest to the heavier mass. If the two masses are equal the center of mass will be in the center. See Figure Position Of Center of Mass and figure Linear Velocity Of Center of Mass The position of the center of mass can be calculated by.

x = d * ma / (ma + mb)
vc = (va * ma + vb * mb) / (ma + mb)
rearranging
d = (ma + mb) * x / ma
mb = d * ma / x - ma
vb = ((ma + mb) * vc - va * ma) / mb
va = ((ma + mb) * vc - vb * mb) / ma

where

ma mb = mass a and b
va vb = velocity of mass a and b
d = distance between ma and mb
x = distance from mb to the center of mass

If two masses collide and are joined together after the collision, the velocity of the joined masses is equal to the velocity of the center of mass.

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BETA : Translation : Center of Mass 02 : Center Of Mass : Linear Relative Velocity Of Center Of Mass : BETA Physics Homework Exercise

Description : Relative velocity of the center of mass for a linear two mass system.

Discussion : The relative velocity of the center of mass is smaller for the heavier mass. See Figure Linear Velocity Of Center of Mass The relative velocity of the center of mass can be calculated by.

vca = (vb - va) * mb /(ma + mb)
vca = vba * mb /(ma + mb)
vc = vca + va
vba = vb - va
rearranging
vba = vca * (ma + mb) / mb
ma = vba * mb / vca - mb
vca = Vc - va
va = vc - vca
vb = vba + va

where

ma mb = mass a and b
va vb = velocity of mass a and b
vc = velocity of center of mass
vca = velocity of center of mass relative to mass a
vba = velocity of mass b relative to mass a

The velocity of mass a relative to mass b is equal and opposite to the velocity of mass b relative to mass a.

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BETA : Translation : Center of Mass 03 : Center Of Mass : Linear Collision Masses Separate After Collision : BETA Physics Homework Exercise

Description : Velocity of the center of mass after the collision of two masses in a linear system, masses separate after the collision.

Discussion : The velocity of the center of mass is constant before and after the collision. See Figure Linear Collision Of Two Masses Separated After Impact

vc = (vai * ma + vbi * mb) / (ma + mb)
vc = (vaf * ma + vbf * mb) / (ma + mb)
rearranging
vaf = (vc * (ma + mb) - vbf * mb) / ma
vbi = (vc * (ma + mb) - vai * ma) / mb

where

ma mb = mass a and b
vai vbi = initial velocity of masses a and b
vaf vbf = final velocity of masses a and b
vc = velocity of center of mass

This is an alternative method to the conservation of momentum method. See Figure Conservation of Linear Momentum - Masses Separate After Collision

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BETA : Translation : Center of Mass 11 : Center Of Mass : Vector Position Of Center of Mass : BETA Physics Homework Exercise

Description : Position of the center of mass for a two dimensional two mass system.

Discussion : The center of mass is on the line between the two masses, and is closest to the heavier mass. If the two masses are equal the center of mass will be halfway between the two masses. See Figure Vector Velocity Of Center Of Mass For Two Masses The position of the center of mass can be calculated by.

d = √((ax - bx)2 + (ay - by)2)
θd = atan2d(ay - by, ax - bx)
x = d * ma / (ma + mb)
cx = bx + x * cosd(θd)
cy = by + x * sind(θd)

Solving For a

x = √((cx - bx)2 + (cy - by)2)
θd = atan2d(cy - by, cx - bx)
d = x * (ma + mb) / ma
ax = bx + d * cosd(θd)
ay = by + d * sind(θd)

where

ma mb = mass a and b
ax ay bx by cx cy = cartesian coordinates of mass a b and center of mass c positions
d = distance between ma and mb
x = distance from mb to the center of mass c
θd = angle d

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BETA : Translation : Center of Mass 12 : Center Of Mass : Vector Velocity Of Center of Mass : BETA Physics Homework Exercise

Description : Velocity of the center of mass for a two dimensional two mass system.

Discussion : The center of mass is on the line between the two masses, and is closest to the heavier mass. If the two masses are equal the center of mass will be halfway between the two masses. See Figure Vector Velocity Of Center Of Mass For Two Masses The velocity of the center of mass can be calculated by.

vx = (ma * va * cosd(θa) + mb * vb * cosd(θb)) / (ma + mb)
vy = (ma * va * sind(θa) + mb * vb * sind(θb)) / (ma + mb)
vc = √(vx2 + vy2)
θc = atan2d(vy, vx)

Solving For a

vx = vc * cosd(θc)
vy = vc * sind(θc)
user defined variables vax and vay
vax = (vx * (ma + mb) - mb * vb * cosd(θb)) / ma
vay = (vy * (ma + mb) - mb * vb * sind(θb)) / ma
solve for va
va = √(vax2 + vay2)
θa = atan2d(vay, vax)

where

ma mb = mass a and b
va vb vc = velocity of a b and center of mass c
vx vy = cartesian coordinates of vc
vax vay = cartesian coordinates of va

If two masses collide and are joined together after the collision, the velocity of the joined masses is equal to the velocity of the center of mass.

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