John's mass is 92 kg, and Barbara's is 67 kg. He is standing on the x axis at xJ = +7.50 m, while she is standing on the x axis at xB = +2.50 m. They switch positions. How far and in which direction does their center of mass move as a result of the switch?
Answer in distance (m)
To determine the distance and direction their center of mass moves after they switch positions, we need to calculate the center of mass given their masses and positions.
First, let's calculate the initial center of mass before the switch:
The total mass is the sum of John's mass (92 kg) and Barbara's mass (67 kg):
Total mass (M) = Mass of John (mJ) + Mass of Barbara (mB) = 92 kg + 67 kg = 159 kg
The x-coordinate of the initial center of mass (xCM_initial) is calculated using the equation:
xCM_initial = [(mJ * xJ) + (mB * xB)] / (mJ + mB)
Substituting the given values:
xCM_initial = [(92 kg * 7.50 m) + (67 kg * 2.50 m)] / (92 kg + 67 kg)
xCM_initial = (690 kg·m + 167.5 kg·m) / 159 kg
xCM_initial = 857.5 kg·m / 159 kg
xCM_initial ≈ 5.394 m
After they switch positions, John is now at xB = +2.50 m, and Barbara is at xJ = +7.50 m.
Now, let's calculate the final center of mass after the switch:
Using the same equation as before:
xCM_final = [(mJ * xJ) + (mB * xB)] / (mJ + mB)
Substituting the given values:
xCM_final = [(92 kg * 2.50 m) + (67 kg * 7.50 m)] / (92 kg + 67 kg)
xCM_final = (230 kg·m + 502.5 kg·m) / 159 kg
xCM_final = 732.5 kg·m / 159 kg
xCM_final ≈ 4.61 m
Therefore, the center of mass moves a distance of approximately |xCM_final - xCM_initial| = |4.61 m - 5.394 m|.
The distance is approximately 0.784 m.
Since the initial center of mass was to the right of the final center of mass, the direction of the movement is to the left (negative x direction).
Hence, their center of mass moves approximately 0.784 m to the left.
To determine the direction and distance their center of mass moves as a result of the switch, we can use the concept of the center of mass formula. The center of mass of a system is the weighted average position of all the masses in the system.
First, let's calculate the initial position of their center of mass before the switch. We need to find the weighted average position using their masses and positions.
The equation for calculating the center of mass is:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
where x_cm is the position of the center of mass, m1 and m2 are the masses, and x1 and x2 are their respective positions.
Using the given information:
m1 = 92 kg (John's mass)
x1 = +7.50 m (John's initial position)
m2 = 67 kg (Barbara's mass)
x2 = +2.50 m (Barbara's initial position)
Let's substitute these values into the equation for the initial position of the center of mass:
x_cm = (92 kg * 7.50 m + 67 kg * 2.50 m) / (92 kg + 67 kg)
Simplifying the equation:
x_cm = (690 kg*m + 167.5 kg*m) / 159 kg
x_cm = 857.5 kg*m / 159 kg
x_cm ≈ 5.4 m
So, the initial position of their center of mass is approximately at x_cm ≈ 5.4 m.
Now, let's calculate the final position of their center of mass after the switch. Since they switch positions, John will be at xB = +2.50 m, and Barbara will be at xJ = +7.50 m.
Using the same equation for the center of mass:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
Substituting the new values:
m1 = 92 kg (John's mass)
x1 = +2.50 m (Barbara's new position)
m2 = 67 kg (Barbara's mass)
x2 = +7.50 m (John's new position)
x_cm = (92 kg * 2.50 m + 67 kg * 7.50 m) / (92 kg + 67 kg)
Simplifying the equation:
x_cm = (230 kg*m + 502.5 kg*m) / 159 kg
x_cm = 732.5 kg*m / 159 kg
x_cm ≈ 4.6 m
Therefore, after the switch, the center of mass moves approximately 4.6 meters to the left (in the negative x-direction) from its initial position.