Three uniform meter sticks, each of mass M, are placed on the floor as follows: stick 1 lies along the y axis from y = 1.3 to y = 2.3 m, stick 2 lies along the x axis from x = 0.0 to x = 1 m, stick 3 lies along the x axis from x = 1.0 m to x = 2.0 m.
Find the location of the center of mass of the meter sticks.
totalmass*cg=mass1*cm1+mass2*cm2+ ...
cg= 1/3(1.3+.5 +0+.5+1+.5)=1/3(3.8)=..
check my figuring.
To find the location of the center of mass of the meter sticks, you need to calculate the weighted average of their positions. The center of mass is defined as the point where the mass of an object can be considered to be concentrated.
To calculate the center of mass, follow these steps:
1. Assign a coordinate system. In this case, let's use the x-axis for horizontal displacement and the y-axis for vertical displacement.
2. Calculate the position and mass of each individual meter stick.
- For stick 1: The y-coordinate is in the range from 1.3 m to 2.3 m, so its position is at y = (1.3 + 2.3) / 2 = 1.8 m. Since all three sticks have the same mass, the mass of stick 1 is M.
- For stick 2: The x-coordinate is in the range from 0.0 m to 1.0 m, so its position is at x = (0 + 1) / 2 = 0.5 m. The mass of stick 2 is also M.
- For stick 3: The x-coordinate is in the range from 1.0 m to 2.0 m, so its position is at x = (1 + 2) / 2 = 1.5 m. The mass of stick 3 is also M.
3. Calculate the total mass of all the sticks. Since each stick has the same mass M, the total mass is 3M.
4. Calculate the coordinates of the center of mass in both the x and y directions using the weighted average formula:
- For the x-coordinate: (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3)
- For the y-coordinate: (m1 * y1) / (m1 + m2 + m3)
Plugging in the values, we get:
- x-coordinate = (M * 0.5 + M * 1.5) / (3M) = 1.0 m
- y-coordinate = (M * 1.8) / (3M) = 0.6 m
The center of mass of the meter sticks is located at (1.0 m, 0.6 m) on the coordinate plane.