John’s mass is 81.0 kg, and Barbara’s is 62.4 kg. He is standing on the x axis at xJ = +9.41 m, while she is standing on the x axis at xB = +4.34 m. They switch positions. How far and in which direction does their center of mass move as a result of the switch?
Distance moved by center of mass =? m toward the origin
To find the distance and direction in which the center of mass moves as a result of the switch, we need to calculate the position of the center of mass before and after the switch.
The position of the center of mass, x_cm, is given by the formula:
x_cm = (mJ * xJ + mB * xB) / (mJ + mB)
Where:
- x_cm is the position of the center of mass,
- mJ and mB are the masses of John and Barbara, respectively, and
- xJ and xB are their initial positions on the x-axis.
Before the switch:
mJ = 81.0 kg (mass of John)
xB = +4.34 m (initial position of Barbara)
mB = 62.4 kg (mass of Barbara)
xJ = +9.41 m (initial position of John)
Using the formula, we can calculate the position of the center of mass before the switch:
x_cm_before = (81.0 kg * 9.41 m + 62.4 kg * 4.34 m) / (81.0 kg + 62.4 kg)
After the switch, John and Barbara switch positions, so:
mJ = 62.4 kg (mass of John after the switch)
xB = +9.41 m (position of Barbara after the switch)
mB = 81.0 kg (mass of Barbara after the switch)
xJ = +4.34 m (position of John after the switch)
Using the formula again, we can calculate the position of the center of mass after the switch:
x_cm_after = (62.4 kg * 4.34 m + 81.0 kg * 9.41 m) / (62.4 kg + 81.0 kg)
To find the distance and direction in which the center of mass moves, we subtract the initial position from the final position:
Distance moved by center of mass = |x_cm_after - x_cm_before|
Finally, we determine the direction based on the sign of the calculated distance. If the sign is positive, the center of mass has moved toward the origin (in the negative x direction). If the sign is negative, the center of mass has moved away from the origin (in the positive x direction).
Therefore, to find the distance and direction in which the center of mass moves as a result of the switch, you can plug the given values into the formulas above and perform the calculations.