John's mass is 86 kg, and Barbara's is 55 kg. He is standing on the x axis at xj = +9.0 m, while she is standing on the x axis at xb = +2.0 m. They switch positions. How far and in which direction does their center of mass move as a result of the switch?
They key thing you need to know is what the center of mass formula is.
Center of mass:
(m1*x1+m2*x2)/(m1+m2)
So at initial center of mass:
(86*9+55*2)/(86+55) = 6.27 m
So now do apply the same equation when they switch position:
(55*9+86*2)/(86+55) = 4.73 m
So now find the difference in position:
Final - initial position:
4.73 - 6.27 = ????
John's mass is 86 kg, and Barbara's is 55 kg. He is standing on the x axis at xj = +9.0 m, while she is standing on the x axis at xb = +2.0 m. They switch positions. How far and in which direction does their center of mass move as a result of the switch?
To find the center of mass, we need to take into account both the masses and their respective positions. The center of mass is given by the equation:
x_cm = (m1 x1 + m2 x2) / (m1 + m2)
where:
x_cm = center of mass position
m1 = mass of the first person
x1 = position of the first person
m2 = mass of the second person
x2 = position of the second person
Let's calculate the initial position of the center of mass:
x_cm_initial = (m1 x1 + m2 x2) / (m1 + m2)
= (86 kg * 9.0 m + 55 kg * 2.0 m) / (86 kg + 55 kg)
= (774 kg·m + 110 kg·m) / 141 kg
= 884 kg·m / 141 kg
≈ 6.277 m
So, before the switch, the center of mass was located approximately 6.277 meters from the origin in the positive x direction.
Now, let's calculate the final position of the center of mass after they switch positions.
After the switch, John will be at xb = +2.0 m and Barbara will be at xj = +9.0 m.
x_cm_final = (m1 x1 + m2 x2) / (m1 + m2)
= (55 kg * 9.0 m + 86 kg * 2.0 m) / (55 kg + 86 kg)
= (495 kg·m + 172 kg·m) / 141 kg
= 667 kg·m / 141 kg
≈ 4.734 m
Therefore, after the switch, the center of mass moves to a new position approximately 4.734 meters from the origin in the positive x direction.
To find the distance and direction the center of mass moved, we can subtract the final position from the initial position:
distance = |x_cm_final - x_cm_initial|
= |4.734 m - 6.277 m|
= |-1.543 m|
= 1.543 m
So, the center of mass moves a distance of 1.543 meters.
Since the initial position was in the positive x direction and the final position is smaller than the initial position, the center of mass moved in the negative x direction.
Therefore, the center of mass moved 1.543 meters in the negative x direction as a result of the switch.