Question

Given: A flat dance floor of dimensions ℓx = 20 m by ℓy = 17 m and has a mass of M = 1100 kg. Use the bottom left corner of the dance floor as the origin. Three dance couples, each of mass m = 100 kg start in the top left, top right, and bottom left corners. What is the initial y coordinate of the center of gravity of the dance floor and three couples? Answer in units of m.

The couple in the bottom left corner moves ℓδ = 8.9 m to the right. What is the new x coordinate of the center of gravity? Answer in units of m.

What was the speed of the center of gravity if it took that couple 6.4 s to change positions? Answer in units of m/s.

Answer #1

y_{df} = y- coordinate of center of dance floor = 17/2 =
8.5

y_{ll} = y- coordinate of lower left =
0 ,

y_{ur} = y- coordinate of upper right = 17

y_{tr} = y- coordinate of top right = 17

center of gravity of the system is given as :

y_{cm} = 2m (y_{ll}) + 2m (y_{ur} ) + 2m
(y_{tr} ) + M (y_{df} ) / (m + m + m + M)

y_{cm} = 200 x 0 + 200 x 17 + 200 x 17 + 1100 x 8.5 /
(200 + 200 + 200 + 1100)

y_{cm} = 9.5

X-coordinate before the bottom left couple move :

X_{df} = X- coordinate of center of dance floor = 20/2 =
10

X_{ll} = X- coordinate of lower left = 0
,

X_{ur} = X- coordinate of upper right = 20

X_{tl} = X- coordinate of top left = 0

center of gravity of the system is given as :

X_{cm} = 2m (X_{ll}) + 2m (X_{ur} ) + 2m
(y_{tl} ) + M (X_{df} ) / (m + m + m + M)

X_{cm} = 200 x 0 + 200 x 20 + 200 x 0 + 1100 x 10 / (200
+ 200 + 200 + 1100)

X_{cm} = 8.82 m

After the bottom left coubple moves to X = 8.9 m

X_{df} = X- coordinate of center of dance floor = 20/2 =
10

X_{ll} = X- coordinate of lower left = 8.9
,

X_{ur} = X- coordinate of upper right = 20

X_{tl} = X- coordinate of top left = 0

center of gravity of the system is given as :

X_{cm} = 2m (X_{ll}) + 2m (X_{ur} ) + 2m
(y_{tl} ) + M (X_{df} ) / (m + m + m + M)

X_{cm} = 200 x 8.9 + 200 x 20 + 200 x 0 + 1100 x 10 /
(200 + 200 + 200 + 1100)

X_{cm} = 9.87 m

distance moved along X-direction :

d = 9.87 - 8.82 = 1.05 m

t = time taken = 6.4 s

speed = distance / time = 1.05 / 6.4 = 0.164 m/s

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