Given: A flat dance floor of dimensions lx = 24 m by ly =23m and has a mass of M = 1200 kg. Use the bottom left corner of the dance floor as the origin. Three dance couples, each of mass m = 110 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.
To find the initial y coordinate of the center of gravity of the dance floor and three couples, we need to calculate the total mass and the total mass moment about the y-axis.
First, let's calculate the total mass (M_total) by summing the individual masses of the dance floor and three couples:
M_total = M + 3m
Given:
M = 1200 kg (mass of the dance floor)
m = 110 kg (mass of each couple)
Substituting the values, we have:
M_total = 1200 kg + 3 * 110 kg
M_total = 1200 kg + 330 kg
M_total = 1530 kg
Next, let's calculate the mass moment about the y-axis (M_y) by finding the sum of moments of the dance floor and three couples.
Moment of the dance floor (M_floor):
The moment of the dance floor is given by the product of its mass (M) and the y-coordinate of its center of gravity (y_floor). Since the dance floor is symmetrical along the y-axis (width is 23m), the center of gravity will be at the midpoint.
y_floor = ly / 2 = 23m / 2 = 11.5m
M_floor = M * y_floor
M_floor = 1200 kg * 11.5m
M_floor = 13800 kg*m
Moment of the couples (M_couples):
The moments of the three couples can be found by multiplying the mass of each couple (m) by their respective y-coordinates.
Couple 1 (top left corner):
y_couple1 = lx / 2 = 24m / 2 = 12m
M_couple1 = m * y_couple1
M_couple1 = 110 kg * 12m
M_couple1 = 1320 kg*m
Couple 2 (top right corner):
y_couple2 = lx / 2 = 24m / 2 = 12m
M_couple2 = m * y_couple2
M_couple2 = 110 kg * 12m
M_couple2 = 1320 kg*m
Couple 3 (bottom left corner):
y_couple3 = 0m (since the bottom left corner is the origin)
M_couple3 = m * y_couple3
M_couple3 = 110 kg * 0m
M_couple3 = 0 kg*m
Now, let's calculate the total mass moment about the y-axis (M_y) by summing the moments:
M_y = M_floor + M_couple1 + M_couple2 + M_couple3
M_y = 13800 kg*m + 1320 kg*m + 1320 kg*m + 0 kg*m
M_y = 16440 kg*m
Finally, we can find the initial y-coordinate of the center of gravity (y_cg) using the formula:
y_cg = M_y / M_total
y_cg = 16440 kg*m / 1530 kg
y_cg ≈ 10.75 m
Therefore, the initial y-coordinate of the center of gravity of the dance floor and three couples is approximately 10.75 m.