consider 4 masses mA=1kg, mB=2kg, mC=3kg, mD=4kg, placed at the vertices of a square ABCD of side 4cm. Specify the position of G the center of mass of the system{A,B,C,D}.
determine the position of G.
Thank you :)
well it is easy.
consider CMx:
CMx*totalmass=mA*locationA+ massB*locationB+ ...
CMx*10kg=1kg*0+2kg*4cm+3kg*4cm+4kg*0
CMx= 20cm/10= 2cm
CMy*10kg= 1kg*0+2kg*0+ 3kg*4cm+4kg*4cm
Cmy= 2.8cm
check my work.
I located the square with A at the origin, B at 4,0, C at 4,4, and D at 0,4
To determine the position of the center of mass, G, of the system {A, B, C, D}, we need to find the coordinates (x, y) of G in relation to a reference point. In this case, let's consider vertex A as the reference point.
Step 1: Calculate the total mass of the system.
The total mass, M, of the system is the sum of the individual masses: mA + mB + mC + mD.
M = 1kg + 2kg + 3kg + 4kg
M = 10kg
Step 2: Find the x-coordinate of G.
To find the x-coordinate of G, we need to take the weighted average of the x-coordinates of each mass, using their respective masses as weights.
xG = (mAx + mBx + mCx + mDx) / M
Given that the side of the square is 4cm, the x-coordinates of A, B, C, and D are:
xA = 0cm
xB = 4cm
xC = 4cm
xD = 0cm
Substituting the values into the formula:
xG = (1kg * 0cm + 2kg * 4cm + 3kg * 4cm + 4kg * 0cm) / 10kg
xG = (0cm + 8cm + 12cm + 0cm) / 10kg
xG = 20cm / 10kg
xG = 2cm
Step 3: Find the y-coordinate of G.
Similarly, to find the y-coordinate of G, we take the weighted average of the y-coordinates of each mass, using their respective masses as weights.
yG = (mAy + mBy + mCy + mDy) / M
Using the same coordinates as before, the y-coordinates of A, B, C, and D are:
yA = 0cm
yB = 0cm
yC = 4cm
yD = 4cm
Substituting the values into the formula:
yG = (1kg * 0cm + 2kg * 0cm + 3kg * 4cm + 4kg * 4cm) / 10kg
yG = (0cm + 0cm + 12cm + 16cm) / 10kg
yG = 28cm / 10kg
yG = 2.8cm
So, the position of the center of mass G is at coordinates (xG, yG), which is (2cm, 2.8cm) relative to vertex A.