Four point masses (A through D) are arranged in a square as shown; the square is

Question

Four point masses (A through D) are arranged in a square as shown; the square is 1.00 m on a side. Where is the center of mass of this array of masses if (A) mA = 1kg , mB = 2 kg , mC = 4kg , mD = 3 kg ; (B) mA = 1 kg , mB = 2 kg , mC = 10 kg , mD = 20 kg . (C) Assume the same masses as in part (A), but now assume that all the particles are moving at the same speed, 2m/s, each directed toward the geometric center of the square. With what velocity is the center of mass moving? If it is not moving at all, say so and explain why. (D) Same as part (C), but using the masses specified in part (B). (E) Suppose the moving particles of part (C) collide completely inelastically and merge. What will be the velocity of the consolidated mass?
A B D

Explanation / Answer

A]

Here from the figure we have  mA = 1kg , mB = 2 kg , mC = 4kg , mD = 3 kg

A is at the point [x1,y1]= [0,1]

B  is at the point [x2,y2]= [1,1]

C  is at the point [x3,y3]= [0,0]

D  is at the point [x4,y4]= [1,0]

X= [ mAx1+mBx2+mCx3+mDx4 ] / [ mA+mB+mC+mD ]

X= [1*0+2*1+4*0+3*1] / [ 1+2+4+3]= [2+3] / 10= 5/10= 1/2=0.5

Y= [ mAy1+mBy2+mCy3+mDy4 ] / [ mA+mB+mC+mD ]

Y= [1*1+2*1+4*0+3*0] / [ 1+2+4+3]= [1+2] / 10= 3/10= 0.3

Center of mass[ X, Y]= [0.5,0.3]m

B]

Here we have  mA = 1kg , mB = 2 kg , mC = 10kg , mD = 20 kg

X= [ mAx1+mBx2+mCx3+mDx4 ] / [ mA+mB+mC+mD ]

Y= [1*1+2*1+10*0+20*0] / [ 1+2+10+20]= [1+2] / 33= 3/33= 0.09

X= [1*0+2*1+10*0+20*1] / [ 1+2+10+20]= [22] / 33= 0.67

Y= [ mAy1+mBy2+mCy3+mDy4 ] / [ mA+mB+mC+mD ]

Center of mass[ X, Y]= [0.67,0.09]m

C]

we have  mA = 1kg , mB = 2 kg , mC = 4kg , mD = 3 kg

Vx= [ mAvx1+mBvx2+mCvx2+mDvx4 ] / [ mA+mB+mC+mD ]

Vx= [ 1*2+2*2+4*2+3*2 ] / [ 1+2+4+3] = [2+4+8+6] / 10= 20/10= 2 m/s

Vy= [ mAvy1+mBvy2+mCvy2+mDvy4 ] / [ mA+mB+mC+mD ]

Vy= [ 1*2+2*2+4*2+3*2 ] / [ 1+2+4+3] = [2+4+8+6] / 10= 20/10= 2 m/s

V cm = [ Vx, Vy] = [2m/s, 2 m/s]

D]

mA = 1kg , mB = 2 kg , mC = 10kg , mD = 20 kg

Vx= [ mAvx1+mBvx2+mCvx2+mDvx4 ] / [ mA+mB+mC+mD ]

Vx= [ 1*2+2*2+10*2+20*2 ] / [ 1+2+4+3] = [2+4+20+40] / 10= 66/10= 6.6m/s

Vy= [ mAvy1+mBvy2+mCvy2+mDvy4 ] / [ mA+mB+mC+mD ]

Vy= [ 1*2+2*2+10*2+20*2 ] / [ 1+2+4+3] = [2+4+20+40] / 10= 66/10= 6.6m/s

V cm = [ Vx, Vy] = [6.6m/s, 6.6 m/s]

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