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 (y-coord CG) of the dance floor and three couples, we need to calculate the total mass and the moment about the y-axis.
Let's break down the problem step by step:
1. Calculate the total mass (M_total) of the dance floor and three couples:
- Dance floor mass (M_floor): Given as M = 1200 kg.
- Mass of three couples (M_couples): Each couple has a mass of m = 110 kg.
- Total mass: M_total = M_floor + M_couples.
M_total = 1200 kg + 3 * 110 kg.
2. Calculate the sum of moments about the y-axis (sum of M*y):
- Dance floor moment (M_floor * y_floor): Since the dance floor is symmetric, its center of gravity (CG_floor) will be at the middle of the floor, which corresponds to half the length of y. Thus, y_floor = ly / 2 = 23 m / 2.
- Couples' moments (3 * M_couples * y_couples): Each couple starts at a corner, so their y-coordinates are the same as the corners they occupy.
- Top left couple (y_1): Since they start at the top left corner, their y-coordinate is y_1 = ly.
- Top right couple (y_2): Since they start at the top right corner, their y-coordinate is y_2 = ly.
- Bottom left couple (y_3): Since they start at the bottom left corner, their y-coordinate is y_3 = 0.
- Sum of moments: sum_My = M_floor * y_floor + 3 * (M_couples * y_couples).
sum_My = (M_floor * y_floor) + 3 * (M_couples * (y_1 + y_2 + y_3)).
3. Calculate the y-coordinate of the center of gravity (CG):
- CG_y = sum_My / M_total.
Now we can substitute the given values into the equations and solve for the initial y-coordinate of the center of gravity (CG_y).