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