John’s mass is 82.7 kg, and Barbara’s is 61.9 kg. He is standing on the x axis at xJ = +9.82 m, while she is standing on the x axis at xB = +4.83 m. They switch positions. How far and in which direction does their center of mass move as a result of the switch?
To determine how far and in which direction the center of mass moves as a result of the switch, we need to calculate the coordinates of the center of mass before and after the switch.
The center of mass (x_CM) of a system of two particles can be calculated using the formula:
x_CM = (m1 * x1 + m2 * x2) / (m1 + m2)
Where:
- x_CM is the x-coordinate of the center of mass
- m1, m2 are the masses of the particles
- x1, x2 are the x-coordinates of the particles
Let's calculate the initial x_CM:
x_CM_initial = (m1 * xJ + m2 * xB) / (m1 + m2)
= (82.7 kg * 9.82 m + 61.9 kg * 4.83 m) / (82.7 kg + 61.9 kg)
= 805.1049 kg·m / 144.6 kg
= 5.5729 m
The positive value indicates that the center of mass is to the right of the origin, along the positive x-axis.
After the switch, John and Barbara exchange positions. Their new x-coordinates become xB' = +9.82 m and xJ' = +4.83 m, respectively.
Using the same formula, we can calculate the final x_CM:
x_CM_final = (m1 * xJ' + m2 * xB') / (m1 + m2)
= (82.7 kg * 4.83 m + 61.9 kg * 9.82 m) / (82.7 kg + 61.9 kg)
= 801.0741 kg·m / 144.6 kg
= 5.5320 m
Comparing the initial and final x_CM values, we find:
Δx_CM = x_CM_final - x_CM_initial
= 5.5320 m - 5.5729 m
= -0.0409 m
The negative value indicates that the center of mass moves in the negative x-direction, which means it moves to the left.
Therefore, the center of mass moves a distance of 0.0409 meters to the left as a result of the switch.